Deformometer for determining deformation of an optical cavity optic

ABSTRACT

A deformometer includes: a cavity body; entry and exit optical cavity optics, such that the optical cavity produces filtered combined light from combined light; a first laser that provides first light; a second laser that provides second light; an optical combiner that: receives the first light; receives the second light; combines the first light and the second light; produces combined light from the first light and the second light; and communicates the combined light to the entry optical cavity optic; a beam splitter that: receives the filtered combined light; splits the filtered combined light; a first light detector in optical communication with the beam splitter and that: receives the first filtered light from the beam splitter; and produces a first cavity signal from the first filtered light; and a second light detector that: receives the second filtered light; and produces a second cavity signal from the second filtered light.

CROSS REFERENCE TO RELATED APPLICATIONS

The application claims priority to U.S. Provisional Patent ApplicationSer. No. 62/714,953 filed Aug. 6, 2018, the disclosure of which isincorporated herein by reference in its entirety.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH

This invention was made with United States Government support from theNational Institute of Standards and Technology (NIST), an agency of theUnited States Department of Commerce. The Government has certain rightsin the invention. Licensing inquiries may be directed to the TechnologyPartnerships Office, NIST, Gaithersburg, Md., 20899; voice (301)301-975-2573; email tpo@nist.gov; reference Ser. No. 16/520,687.

BRIEF DESCRIPTION

Disclosed is a deformometer for determining deformation of an opticalcavity disposed on an optical cavity, the deformometer comprising: theoptical cavity comprising: a cavity body; an entry optical cavity opticdisposed at an entry end of cavity body and that receives combinedlight; and an exit optical cavity optic disposed at an exit end ofcavity body, the entry optical cavity optic in optical communication andoptically opposing the exit optical cavity optic, such that the exitoptical cavity optic receives the combined light from the entry opticalcavity optic, such that the optical cavity produces filtered combinedlight from the combined light; a first laser in optical communicationwith entry optical cavity optic and that provides first light; a secondlaser in optical communication with entry optical cavity optic and thatprovides second light; an optical combiner in optical communication withthe entry optical cavity optic and that: receives the first light fromthe first laser; receives the second light from the second laser;combines the first light and the second light; produces combined lightfrom the first light and the second light; and communicates the combinedlight to the entry optical cavity optic; a beam splitter in opticalcommunication with the exit optical cavity optic and that: receives thefiltered combined light from the optical cavity; splits the filteredcombined light into first filtered light and second filtered light; afirst light detector in optical communication with the beam splitter andthat: receives the first filtered light from the beam splitter; andproduces a first cavity signal from the first filtered light; and asecond light detector in optical communication with the beam splitterand that: receives the second filtered light from the beam splitter; andproduces a second cavity signal from the second filtered light, fromwhich a deformation of the entry optical cavity optic and exit opticalcavity optic is determined.

Disclosed is a deformometer for determining deformation of an opticalcavity optic disposed on an optical cavity, the deformometer comprising:the optical cavity comprising: a cavity body; an entry optical cavityoptic disposed at an entry end of cavity body and that receives combinedlight; and an exit optical cavity optic disposed at an exit end ofcavity body, the entry optical cavity optic in optical communication andoptically opposing the exit optical cavity optic, such that the exitoptical cavity optic receives the combined light from the entry opticalcavity optic, such that the optical cavity produces filtered combinedlight from the combined light; a first laser in optical communicationwith entry optical cavity optic and that provides first light; a secondlaser in optical communication with entry optical cavity optic and thatprovides second light; a first optical combiner in optical communicationwith the entry optical cavity optic and that: receives the first lightfrom the first laser; receives the second light from the second laser;combines the first light and the second light; produces combined lightfrom the first light and the second light; and communicates the combinedlight to the entry optical cavity optic; a second optical combiner inoptical communication with the exit optical cavity optic and that:receives the filtered combined light from the optical cavity; splits thefiltered combined light into first filtered light and second filteredlight; a first light detector in optical communication with the secondoptical combiner and that: receives the first filtered light from thesecond optical combiner; and produces a first cavity signal from thefirst filtered light; and a beam splitter in optical communication withthe second optical combiner and that: receives the second filtered lightfrom the second optical combiner; communicates a portion of the secondfiltered light to an imager; and communicates a second portion of thesecond filtered light to a second light detector; the second lightdetector in optical communication with the beam splitter and that:receives the second filtered light from the beam splitter; and producesa second cavity signal from the second filtered light, from which adeformation of the entry optical cavity optic and exit optical cavityoptic is determined; and the imager in optical communication with thebeam splitter and that: receives the second filtered light from the beamsplitter; and produces an image signal from the second filtered light;an optical frequency comb source that produces a set of opticalfrequencies; a seed light detector that: receives the comb seed lightfrom the seed laser; receives the optical frequency comb from theoptical frequency comb source; and produces a first reference signalfrom the first light and the first optical frequency comb; and a secondreference light detector that: receives the second light from the secondlaser; receives the second optical frequency comb from the opticalfrequency comb source; and produces a second reference signal from thesecond light and the second optical frequency comb.

Disclosed is a deformometer for determining deformation of an opticalcavity optic disposed on an optical cavity, the deformometer comprising:a first optical cavity comprising: a first cavity body; a first entryoptical cavity optic disposed at an entry end of first cavity body andthat receives optical frequency comb; and a first exit optical cavityoptic disposed at an exit end of first cavity body, the first entryoptical cavity optic in optical communication and optically opposing thefirst exit optical cavity optic, such that the first exit optical cavityoptic receives the combined light from the first entry optical cavityoptic, such that the first optical cavity produces filtered frequencylight from the combined light; an optical frequency comb source inoptical communication with first entry optical cavity optic and thatprovides an optical frequency comb; a beam splitter in opticalcommunication with the first entry optical cavity optic and that:receives the optical frequency comb from the optical frequency combsource; splits a portion of the optical frequency comb to producefeedback light; and communicates the feedback light to optical frequencycomb source as feedback control for the optical frequency comb source; aFourier transform spectrometer in optical communication with the opticalcavity and that: receives the shifted frequency light from the opticalcavity; and produces a deformation signal from the shifted frequencylight from which a deformation of the optical cavity optic and firstexit optical cavity optic is determined.

Disclosed is a deformometer that determines deformation of an opticalcavity optic disposed on an optical cell and includes an optical cellincluding a cell body; an entry optical cell optic disposed at an entryend of cell body and that receives combined light; and an exit opticalcavity optic disposed at an exit end of the cell body, wherein the entryoptical cavity optic is in optical communication and optically opposesexit optical cavity optic, such that the exit optical cell opticreceives the combined light from the entry optical cavity optic, and theoptical cell produces shifted combined light from the combined light; afirst laser in optical communication with entry the entry optical cavityoptic and that provides first light; a second laser in opticalcommunication with the entry optical cavity optic and that providessecond light; a propagation coupler in optical communication with thefirst laser; a beam splitter that receives the first light and thesecond light before communication into the optical cell; a second beamsplitter to receive filtered combined light and shifted combined lightfrom the optical cell; an optical combiner that splits filtered lightreceived from the second beam splitter and produces a first cavitysignal and a second cavity signal; a first light detector that receivesthe first cavity signal and produces a second filtered light; a secondphase detector that receives the second filtered light from the firstlight detector; a second light detector that receives the second cavitysignal from the optical combiner and produces a first cavity signal; anda first phase detector that receives the first cavity signal from thesecond light detector.

Disclosed is a deformometer for determining deformation of an opticalcavity optic disposed on an optical cavity, the deformometer comprising:a first optical cavity comprising: a cavity body; a first entry opticalcavity optic disposed at an entry end of the cavity body; and a firstexit optical cavity optic disposed at an exit end of cavity body, thefirst entry optical cavity optic in optical communication and opticallyopposing the first exit optical cavity optic, such that the firstoptical cavity: receives a reference gas at a first pressure P1;receives a first light; and produces a first filtered light from thefirst light; the first entry optical cavity optic in opticalcommunication and optically opposing the first exit optical cavityoptic, such that the first optical cavity: receives a differentreference gas at a first pressure P1; receives a first light; andproduces a first filtered light from the first light; a second opticalcavity comprising: a second entry optical cavity optic disposed at theentry end of the cavity body; and a second exit optical cavity opticdisposed at the exit end of the cavity body, the second entry opticalcavity optic in optical communication and optically opposing the secondexit optical cavity optic, such that the second optical cavity: receivesa second gas at a second pressure P2; receives a second light; andproduces a second filtered light from the second light; a gas source influid communication with the second optical cavity and providing thesecond gas to the second optical cavity; and a second pump in fluidcommunication with the optical cavity to pump the second gas from theoptical cavity; a first laser in optical communication with the firstoptical cavity and that provides the first light to the first opticalcavity; and a second laser in optical communication with the secondoptical cavity and that provides the second light to the second opticalcavity, such that the deformation of the first optical cavity optic, thesecond optical cavity optic, the third optical cavity optic, and thefourth optical cavity optic is determined from the first cavity signaland the second cavity signal.

Disclosed is a process for determining deformation of an optical cavityoptic disposed on an optical cavity with a deformometer, the processcomprising: combining first light with a second light; producingcombined light from the first light and the second light; receiving, anentry optical cavity optic disposed at an entry end of a cavity body ofa deformometer, the combined light; transmitting, from the first opticalcavity optic to an exit optical cavity optic disposed at an exit end ofthe cavity body, the combined light, the entry optical cavity opticbeing in optical communication and optically opposing the exit opticalcavity optic; receiving, by the exit optical cavity optic, the combinedlight; producing a filtered combined light from the combined light bytransmitted the combined light from the first optical cavity optic andfrom the second optical cavity optic; producing, from the filteredcombined light, a first filtered light and a second filtered light;analyzing the first filtered light and the second filtered light todetermine the deformation of the entry optical cavity optic and the exitoptical cavity optic.

BRIEF DESCRIPTION OF THE DRAWINGS

The following description should not be considered limiting in any way.With reference to the accompanying drawings, like elements are numberedalike.

FIG. 1 shows a deformometer for determining deformation of an opticalcavity optic;

FIG. 2 shows a deformometer for determining deformation of an opticalcavity optic;

FIG. 3 shows a deformometer for determining deformation of an opticalcavity optic;

FIG. 4 shows a deformometer for determining deformation of an opticalcavity optic;

FIG. 5 shows a deformometer for determining deformation of an opticalcavity optic;

FIG. 6 shows a deformometer for determining deformation of an opticalcavity optic;

FIG. 7 shows a deformometer for determining deformation of an opticalcavity optic;

FIG. 8 shows an optical cavity;

FIG. 9 shows a refractometer in panel A and panel B and a deformation inpanel C for cavity lengths per pascal of pressure on the measurementcavity when the reference cavity is at vacuum;

FIG. 10 shows a graph of a disagreement in pressure as measured by twoseparate laser refractometers (pFP) and mercury ultrasonic manometer(pUIM);

FIG. 11 shows a correction to an optical cavity for distortion viafinite-element analysis and an inspection of the mode position on themirror. Panel (a) is an image of the mirror showing the bond interface.Through edge-detection, an estimate can be made of the area upon whichthe pressure acts. In (b), another image is taken with a laser beamaligned to the cavity resonance. By combining these two images, anestimate of the location of the beam on the mirror surface is made. Theresult of a finite-element analysis is shown in (c) datasheet valueswere used for elastic properties of ULE glass, and the geometry wasestimated by the bond line in panel (a). The difference in mirrorbending calculated by finite-element is extracted as a profile, shown inpanel (d);

FIG. 12 shows (a) MIRE apparatus and (b) refractometry cells of threedifferent lengths but which are otherwise nominally identical. Eachborehole has a gas inlet and outlet;

FIG. 13 shows an refractometry apparatus with a feedback loop fromdetector (Det 1) to keep a fixed wavelength laser locked to atransmission maximum of the FPI. Beam splitters (BS) combine beams fromtwo lasers on Det 2, and the frequency difference is measured by thecounter. “Iso” is a Faraday isolator;

FIG. 14 shows a cavity;

FIG. 15 shows a graph of change in length versus pressure of helium;

FIG. 16 shows a graph of change in beat frequency versus pressure;

FIG. 17 shows a graph of change in length versus temperature;

FIG. 18 shows a graph of beat frequency versus time; and

FIG. 19 shows a graph of interactive beat frequency versus time.

DETAILED DESCRIPTION

A detailed description of one or more embodiments is presented herein byway of exemplification and not limitation.

It has been discovered that a deformometer and process described hereinincludes a plurality of wavelengths of light or species of gas tomeasure a deformation of an optical cavity, cell, or other opticalelement that holds gas, wherein the deformation occurs due to a forceexerted by the gas. The deformometer determines a difference between anindex of refraction of gases, wavelengths, or a combination thereof.Advantageously, a combination of such determines a distortion of anoptical cavity filled with gas.

In an embodiment, with reference to FIG. 1, deformometer 200 determinesdeformation of optical cavity optic 210 disposed on optical cavity 212and includes optical cavity 212 including cavity body 214; entry opticalcavity optic 210.1 disposed at entry end 216 of cavity body 214 and thatreceives combined light 240; and exit optical cavity optic 210.2disposed at exit end 218 of cavity body 214, entry optical cavity optic210.1 in optical communication and optically opposing exit opticalcavity optic 210.2, such that exit optical cavity optic 210.2 receivescombined light 240 from entry optical cavity optic 210.1, and opticalcavity 212 produces filtered combined light 224 from combined light 240;first laser 220.1 in optical communication with entry optical cavityoptic 210.1 and that provides first light 222.1; second laser 220.2 inoptical communication with entry optical cavity optic 210.1 and thatprovides second light 222.2; optical combiner 230 in opticalcommunication with entry optical cavity optic 210.1 and that: receivesfirst light 222.1 from first laser 220.1; receives second light 222.2from second laser 220.2; combines first light 222.1 and second light222.2; produces combined light 240 from first light 222.1 and secondlight 222.2; and communicates combined light 240 to entry optical cavityoptic 210.1; beam splitter 236 in optical communication with exitoptical cavity optic 210.2 and that: receives filtered combined light224 from optical cavity 212; splits filtered combined light 224 intofirst filtered light 226.1 and second filtered light 226.2; first lightdetector 238.1 in optical communication with beam splitter 236 and that:receives first filtered light 226.1 from beam splitter 236; and producesfirst cavity signal 228.1 from first filtered light 226.1; and secondlight detector 238.2 in optical communication with beam splitter 236 andthat: receives second filtered light 226.2 from beam splitter 236; andproduces second cavity signal 228.2 from second filtered light 226.2,from which a deformation of entry optical cavity optic 210.1 and exitoptical cavity optic 210.2 is determined. Deformometer 200 can includelens 232 in optical communication with optical combiner 230 and thatreceives combined light 240 from optical combiner 230 and focuses thecombined light 240. Deformometer 200 can include mirror 234 in opticalcommunication with first laser 220.1 and second laser 220.2. It shouldbe appreciated that although reference is made to determination ofdeformation of optical cavity optic 210 disposed on optical cavity 212,determination of deformation also includes determination of deformationof elements disposed on optical cavity optic 210 such as optical cavity212.

In an embodiment, with reference to FIG. 2, deformometer 200 includesoptical cavity 212. Optical cavity 212 includes cavity body 214; entryoptical cavity optic 210.1 disposed at entry end 216 of cavity body 214and that receives combined light 240; and exit optical cavity optic210.2 disposed at exit end 218 of cavity body 214, wherein entry opticalcavity optic 210.1 is in optical communication and optically opposesexit optical cavity optic 210.2. Exit optical cavity optic 210.2receives combined light 240 from entry optical cavity optic 210.1, andoptical cavity 212 produces filtered combined light 224 from combinedlight 240. First laser 220.1 is in optical communication with entryoptical cavity optic 210.1 and provides first light 222.1. Second laser220.2 is in optical communication with entry optical cavity optic 210.1and provides second light 222.2. Deformometer 200 also includes firstoptical combiner 230.1 in optical communication with entry opticalcavity optic 210.1 and that: receives first light 222.1 from first laser220.1; receives second light 222.2 from second laser 220.2; combinesfirst light 222.1 and second light 222.2; produces combined light 240from first light 222.1 and second light 222.2; and communicates combinedlight 240 to entry optical cavity optic 210.1. Second optical combiner230.2 is in optical communication with exit optical cavity optic 210.2and receives filtered combined light 224 from optical cavity 212; splitsfiltered combined light 224 into first filtered light 226.1 and secondfiltered light 226.2. First light detector 238.1 is in opticalcommunication with second optical combiner 230.2 and receives firstfiltered light 226.1 from second optical combiner 230.2; and producesfirst cavity signal 228.1 from first filtered light 226.1. Beam splitter236 is in optical communication with second optical combiner 230.2 andreceives second filtered light 226.2 from second optical combiner 230.2;communicates a portion of second filtered light 226.2 to imager 280; andcommunicates a second portion of second filtered light 226.2 to secondlight detector 238.2. Second light detector 238.2 is in opticalcommunication with beam splitter 236 and receives second filtered light226.2 from beam splitter 236; and produces second cavity signal 228.2from second filtered light 226.2, from which a deformation of entryoptical cavity optic 210.1 and exit optical cavity optic 210.2 isdetermined. Imager 280 is in optical communication with beam splitter236 and receives second filtered light 226.2 from beam splitter 236; andproduces image signal 282 from second filtered light 226.2. Deformometer200 also includes optical frequency comb source 242 that produces firstoptical frequency comb 244.1 and second optical frequency comb 244.2;first reference light detector 238.3 that: receives first light 222.1from first laser 220.1, receives first optical frequency comb 244.1 fromoptical frequency comb source 242, and produces first reference signal402.1 from first light 222.1 and first optical frequency comb 244.11;and second reference light detector 238.3 that: receives second light222.2 from the second laser 220.2, receives second optical frequencycomb 244.2 from optical frequency comb source 242, and produces secondreference signal 402.2 from second light 222.2 and second opticalfrequency comb 244.2. Deformometer 200 can include probe light detector238.5 in optical communication with first laser 220.1 to receive firstlight 222.1 from first laser 220.1 and to produce probe signal 400 fromfirst light 222.1. Optical isolator 256 can be in optical communicationwith laser 220.1 to optically isolate laser 220.1. Fiber coupler 258 canbe included to optically split or couple multiple laser light beams,e.g., light 222.1, light 222.2, and the like. Coupling of propagation oflight between free space and in fiber optics is provided by propagationcoupler 272. Polarization filtering and control is provided by waveplate248, polarizer 249, and the like. Collimation or focusing of light isprovided by lens 232. Light is directed by mirror 234 and beam splitter236. Filtering of light is provided by filter 270. Detection of lightand conversion of light from an optical domain to an electrical domainis provided by light detector 238 and imager 280.

In an embodiment, with reference to FIG. 3, deformometer 200 includeselectrooptic modulator 276 in optical communication with first laser220.1 that: receives first light 222.1 from first laser 220.1; receivesoscillator signal 302 from oscillator 300; and modulates first light222.1 at a frequency of oscillator signal 302. Electrooptic modulator276 can be optically interposed between first laser 220.1 and firstoptical combiner 230.1. Amplitude optical modulator 290 can be inoptical communication with second laser 220.2 to receive second light222.2 from second laser 220.1 and to modulate second light 222.2.Amplitude optical modulator 290 can be optically interposed betweensecond laser 220.2 and first optical combiner 230.1. It is contemplatedthat light can propagate through free space or in a condensed opticalmedium such as fiber optic 274.

In an embodiment, with reference to FIG. 4, deformometer 200 includessecond optical cavity 212.2 that includes second entry optical cavityoptic 210.3 disposed at entry end 216 of cavity body 214 and thatreceives second combined light 240.2; and second exit optical cavityoptic 210.4 disposed at exit end 218 of cavity body 214. Second entryoptical cavity optic 210.3 is in optical communication and opticallyopposes second exit optical cavity optic 210.4, wherein second exitoptical cavity optic 210.4 receives second combined light 240.2 fromsecond entry optical cavity optic 210.3, and second optical cavity 212.2produces second filtered combined light 224.2 from second combined light240.2. Third laser 220.3 is in optical communication with second entryoptical cavity optic 210.3 and provides third light 222.3; fourth laser220.4 is in optical communication with second entry optical cavity optic210.3 and provides fourth light 222.4. Third optical combiner 230.3 isin optical communication with second entry optical cavity optic 210.3and receives third light 222.3 from third laser 220.3, receives fourthlight 222.4 from fourth laser 220.4, combines third light 222.3 andfourth light 222.4, produces second combined light 240.2 from thirdlight 222.3 and fourth light 222.4, and communicates second combinedlight 240.2 to second entry optical cavity optic 210.3. Fourth opticalcombiner 230.4 is in optical communication with second exit opticalcavity optic 210.4 and receives second filtered combined light 224.2from optical cavity 212 and splits second filtered combined light 224.2into third shifted light 226.3 and fourth shifted light 226.4. Thirdlight detector 238.7 is in optical communication with fourth opticalcombiner 230.4, receives third shifted light 226.3 from fourth opticalcombiner 230.4, and produces third cavity signal 228.3 from thirdshifted light 226.3. Second beam splitter 236 is in opticalcommunication with fourth optical combiner 230.4, receives fourthshifted light 226.4 from fourth optical combiner 230.4, communicates aportion of fourth shifted light 226.4 to second imager 280, andcommunicates a second portion of fourth shifted light 226.4 to sixthlight detector 238.6. Sixth light detector 238.6 is in opticalcommunication with second beam splitter 236, receives fourth shiftedlight 226.4 from second beam splitter 236, and produces fourth cavitysignal 228.4 from fourth shifted light 226.4, from which a deformationof second entry optical cavity optic 210.2 and second exit opticalcavity optic 210.4 is determined. Second imager 280 is in opticalcommunication with second beam splitter 236, receives fourth shiftedlight 226.4 from second beam splitter 236, and produces second imagesignal 282.2 from fourth shifted light 226.4.

Second probe light detector 238.10 is in optical communication withthird laser 220.3, receives third light 222.3 from third laser 220.3,and produces second probe signal 400.2 from third light 222.3. Secondreference light detector 238.9 is in optical communication with secondlaser 220.2 and fourth laser 220.4, receives second light 222.2 fromsecond laser 220.2, receives fourth light 222.4 from fourth laser 220.4,and produces mixed optical signal 404 from second light 222.2 and fourthlight 222.4. Third probe light detector 238.8 is in opticalcommunication with first laser 220.1 and second laser 220.2, receivesfirst light 222.1 from first laser 220.1, receives third light 222.3from third laser 220.3, and produces third probe signal 400.3 from firstlight 222.1 and third light 222.3.

In an embodiment, with reference to FIG. 5, deformometer 200 includesfirst optical cavity 212.1 that includes first cavity body 214.1; firstentry optical cavity optic 210.1 disposed at entry end 216 of firstcavity body 214.1 and that receives optical frequency comb 244; andfirst exit optical cavity optic 210.2 disposed at exit end 218 of firstcavity body 214.1. First entry optical cavity optic 210.1 is in opticalcommunication and optically opposes first exit optical cavity optic210.2, wherein first exit optical cavity optic 210.2 receives combinedlight 240 from first entry optical cavity optic 210.1, and first opticalcavity 212.1 produces shifted frequency light 246 from combined light240. Optical frequency comb source 242 is in optical communication withfirst entry optical cavity optic 210.1 and provides optical frequencycomb 244. Beam splitter 236 is in optical communication with first entryoptical cavity optic 210.1, receives optical frequency comb 244 fromoptical frequency comb source 242, splits a portion of optical frequencycomb 244 to produce feedback light 252, and communicates feedback light252 to optical frequency comb source 242 as feedback control for opticalfrequency comb source 242. Fourier transform spectrometer 250 is inoptical communication with optical cavity 212, receives shiftedfrequency light 246 from optical cavity 212, and produces deformationsignal 254 from shifted frequency light 246 from which a deformation ofoptical cavity optic 210.1 and first exit optical cavity optic 210.2 isdetermined. Waveplate 248 is in optical communication with opticalcavity 212, optically interposed between optical frequency comb source242 and optical cavity 212, and controls a polarization of opticalfrequency comb 244 received by optical cavity 212.

In an embodiment, with reference to FIG. 6, deformometer 200 includesfirst optical cavity 212.1 that includes cavity body 214; first entryoptical cavity optic 210.1 disposed at entry end 216 of cavity body214.1; and first exit optical cavity optic 210.2 disposed at exit end218 of cavity body 214. First entry optical cavity optic 210.1 is inoptical communication and optically opposes first exit optical cavityoptic 210.2. First optical cavity 212.1 receives reference gas 334 atfirst pressure P1, receives first light 222.1, and produces firstfiltered light 226.1 from first light 222.1. Second optical cavity 212.2includes second entry optical cavity optic 210.3 disposed at entry end216 of cavity body 214; and second exit optical cavity optic 210.4disposed at exit end 218 of cavity body 214. Second entry optical cavityoptic 210.3 is in optical communication and optically opposes secondexit optical cavity optic 210.3. Second optical cavity 212.2 receivessecond gas 332 at second pressure P2, receives second light 222.1, andproduces second filtered light 226.2 from second light 222.2. Gas source330 is in fluid communication with second optical cavity 212.2 andprovides second gas 332 to second optical cavity 212.2. Second pump314.2 is in fluid communication with optical cavity 212.2 to pump secondgas 332 from optical cavity 212.2. First pump 314.1 is in fluidcommunication with first optical cavity 212.1 to obtain a selectedpressure thereof at first pressure P1 via pump stem 350 of opticalcavity 212 connected to first optical cavity 212.1 and that provideswall 322 that bounds flow channel 324 through reference gas 334 flows topump 314.1. First laser 220.1 is in optical communication with firstoptical cavity 212.1 and provides first light 222.1 to first opticalcavity 212.1. Second laser 220.2 is in optical communication with secondoptical cavity 212.2 and provides second light 222.2 to second opticalcavity 212.2. Deformation of first optical cavity optic 210.1, secondoptical cavity optic 210.2, third optical cavity optic 210.3, and fourthoptical cavity optic 210.4 is determined from first cavity signal 228.1and second cavity signal 228.2. Valves 328 throttle and isolate gassource 330 and pump 314.

In an embodiment, with reference to FIG. 7, deformometer 200 determinesdeformation of optical cavity optic 410 disposed on optical cell 412 andincludes optical cell 412 including cell body 414; entry optical celloptic 410.1 disposed at entry end 416 of cell body 414 and that receivescombined light 240; and exit optical cavity optic 410.2 disposed at exitend 418 of cell body 414, entry optical cavity optic 410.1 in opticalcommunication and optically opposing exit optical cavity optic 410.2,such that exit optical cell optic 410.2 receives combined light 240 fromentry optical cavity optic 410.1, and optical cell 414 produces shiftedcombined light 424 from combined light 240; first laser 220.1 in opticalcommunication with entry optical cavity optic 410.1 and that providesfirst light 222.1; second laser 220.2 in optical communication withentry optical cavity optic 410.1 and that provides second light 222.2;propagation coupler 272 in optical communication with first laser 220.1;beam splitter 236 that receives first light 222.1 and second light 222.2before communication into optical cell 412; second beam splitter 236 toreceive filtered combined light 224 and shifted combined light 424 fromoptical cell 412; optical combiner 230 that splits filtered light 426received from second beam splitter 236 and to produce first cavitysignal 428.1 and second cavity signal 428.2; first light detector 238.1that receives first cavity signal 428.1 and produces second filteredlight 226.2; second phase detector 430.2 that receives second filteredlight 226.2 from first light detector 238.1; second light detector 238.2that receives second cavity signal 428.2 from optical combiner 230 andproduces first cavity signal 228.1; and first phase detector 430.1 thatreceives first cavity signal 228.1 from second light detector 238.2.

In deformometer 200, optical cavity optic 210 can include a concave,convex, or flat mirror to provide first entry of light into the cavityand reflection of light to create an optical resonator. Exemplaryoptical cavity optic 210 includes a concave dielectric mirror that isanti-reflective coated on the input and highly reflective on the inside.The reflection of optical cavity optic 210 can be from 0.1 to 1.0,specifically from 0.9 to 1.0, and more specifically from 0.999 to 1.0 ata wavelength from the ultraviolet (UV) to infrared (IR), specifically atany two wavelengths within that range. Moreover, the shapes of cavityoptic 210 provide stable optical resonance with no overlapping modes.

In deformometer 200, cavity body 214 can include a tube to mechanicallydefine the optical cavity and can be rectangular parallelepiped glasswith a smooth cylindrical bore. Moreover, cavity body 214 can have anopening for gas from the surrounding pressure vessel 316 to enter thespace between cavity optic 210. A size (e.g., a longest lineardimension) of cavity body 214 can be from 1 mm to 10 m, specificallyfrom 1 cm to 100 cm, and more specifically from 2 cm to 10 cm. Acoefficient of thermal expansion of cavity body 214 can be from 0 to10⁴, specifically from 0 to 10⁻⁶, and more specifically from 0 to 10⁻⁸.In an embodiment, cavity body 214 is made from ultra-low expansion glassin a rectangular parallelepiped shape, with cylindrical bores on theends to secure cavity optic 210 and a slit along one side to allow forthe penetration of gas.

In deformometer 200, laser 220 (e.g., laser 220.1, laser 220.2, laser220.3, laser 220.4) can be a laser source that produces laser lightwhich is resonant or nearly resonant with the cavity, including a HeNelaser, Ti: Sapphire laser, external-cavity diode lasers, and the like.Exemplary laser 220 includes an infrared external cavity diode laser.Moreover, the laser linewidth can be smaller than the free spectralrange of the cavity defined by cavity optic 210. In an embodiment, laser220 includes a HeNe laser and an infrared external cavity diode laser.

In deformometer 200, light 222 (e.g., light 222.1, light 222.2, light222.3, light 222.4) can include a single-frequency laser light used toprobe the cavity resonances. Exemplary light 222 includes light from aHeNe laser. A wavelength of light 222 can be from UV to IR, specificallyfrom 250 nm to 2 um, and more specifically can be a wavelength definedby cavity optic 210. Moreover, the frequency spectrum of light 222 canhave a single frequency peak. A power of light 222 can be from 1 uW to 1W, specifically from 10 uW to 100 mW, and more specifically from 100 uWto 10 mW. When in optical communication with cavity 212, the wavelengthcan correspond to a transmitted wavelength of cavity 212 and a linewidthsmaller than that of the cavity defined by cavity optic 210. When inoptical communication with optical cell 412, the wavelength cancorrespond to a transmitted wavelength of cell 412, and the linewidthcan be smaller than the speed of light divided by the optical pathlength from the source 220 to detector 238. Exemplary beam waists are 1mm, and duty cycle is one. In an embodiment, light 222.1 and 222.2 havewavelength 1542 nm, 10 mW power, 2 kHz linewidth. In an embodiment,light 222.3 and 222.4 have wavelength 633 nm, 1 mW power, 10 kHzlinewidth.

In deformometer 200, filtered combined light 224 and filtered light 226include light that transmits the cavity.

In deformometer 200, cavity signal 228 can include any signal used todetermine if light 222 has the same frequency as light 224. In anembodiment, e.g., as shown in FIG. 1, cavity signal 228 is the frequencyof the light. In an embodiment, e.g., as shown in FIGS. 2-4, cavitysignal 228 is the intensity or power of the transmitted light.

In deformometer 200, optical combiner 230 can include a dichroic mirror,plate beam splitter, cube beam splitter, or partially reflective mirrorto merge light 222 (e.g., light 222.1, light 222.2, light 222.3, light222.4) or to align the light to travel along the same path. Moreover, inreverse, the optical combiner can split

In deformometer 200, mirror 234 is a mirror that reflects light 222(e.g., light 222.1, light 222.2, light 222.3, light 222.4). Exemplarymirrors include dichroic mirrors, dielectric mirrors or metallicmirrors.

In deformometer 200, beam splitter 235 can include an optic that splitsthe incoming beam power equally, with half of the power continuing alongthe original direction and half of the power travelling orthogonally.Exemplary beam splitter are plate 50/50 beam splitters or 50/50 cubebeam splitters.

In deformometer 200, polarizing beam splitter 236 can include an opticthat splits the incoming beam power according to its polarization state,with orthogonal linear polarizations traveling orthogonally to eachother. Exemplary polarizing beam splitter are polarizing beam splittingcubes and Glan-Laser calcite polarizers.

In deformometer 200, light frequency detector 237 transduces thefrequency or wavelength of filtered light 226.

In deformometer 200, light detector 238 can include a detector thattransduces the power of filtered light 226 into cavity signal 228. Thebandwidth of the photodetectors can be between 0 Hz and 10 GHz,specifically between 0 and the frequency of the free spectral rangedefine by cavity 212. Exemplary light detectors include power meters andphotodiodes, photoreceivers.

In deformometer 200, combined light 240 can include laser light 222, buteach (222.1, 222.2, 222.3, etc.) with its own distinct wavelengths,colors, or frequencies. Exemplary combined light 240 includes light frommultiple single frequency laser sources (220) combined together usingoptical combiners (230) or light from a frequency comb source (242).

In deformometer 200, optical frequency comb source 242 can include alaser source that produces multiple laser light 222 spaced equally infrequency space such that each frequency can probe the cavity. Exemplaryoptical frequency comb sources can be a phase-stabilized femto-secondpulse laser or phase-modulated continuous wave laser light.

In deformometer 200, optical frequency comb 244 can include multiplelaser light 222 spaced equally in frequency space such that eachfrequency can probe the cavity. Moreover, optical frequency comb 244 isa specific example of combined light 240, where the wavelengthcomponents of combined light 240 have equal separation in frequency.

In deformometer 200, filtered frequency comb light 246 is product of theinput light and filter function that is filtered by cavity optics 210.

In deformometer 200, waveplate 248 and polarizer 249 can include anoptic to manipulate a polarization of light 222 and combined light 240.Exemplary waveplates and polarizers include birefringent crystals andcalcite polarizers.

In deformometer 200, Fourier transform spectrometer 250 can include anydevice that determines the transmission each component of the filteredfrequency comb light 244. Exemplars include a Michelson interferometer,or second comb with shifted repetition rate or etalon, diffractiongrating, CCD camera, and the like. An output from the Fourier transformspectrometer 250 is a transmission spectrum of cavity 212.

In deformometer 200, feedback light 252 is a sample of the frequencycomb light for measurement of an absolute frequency of a frequencycomponent of frequency comb 200.

In deformometer 200, optical isolator 256 can include an optic thatprevents the transmission of light in one direction but not the other.Exemplary optical isolators are Faraday effect isolators.

In Deformometer 200, Fiber Coupler 258 can Include an Optic that CouplesLight from Air to a Condensed Matter Medium. The Efficiency of theCoupling can be from 0.1 to 1, Specifically from 0.5 to 1.0. ExemplaryFiber Couplers 258 Include Fiber-to-Free Space Couplers

In deformometer 200, fiber optic 274 is a condensed-matter medium thatpropagates light. Exemplary fiber optics include single-mode fiberoptical cable, polarization-maintaining single-mode fiber optical cable,and the like.

In deformometer 200, fiber optic power splitter 258 is acondensed-matter medium that splits light. Exemplary fiber opticsinclude single-mode fiber optical cable, polarization-maintainingsingle-mode fiber optical cable, and the like.

In deformometer 200, propagation coupler 272 can include an optic thatcouples light from air to a condensed matter medium. The efficiency ofthe coupling can be from 0.1 to 1, and specifically from 0.5 to 1.0.Exemplary fiber couplers include aspheric lenses, achromatic lenses,parabolic mirror couplers, and the like.

In deformometer 200, reference light 260 is a combination of light 222and optical frequency comb 244 used to measure the frequency of light222 relative to frequency comb 244 through beat note detection on highspeed optical detector 238, e.g., a photoreceiver.

In deformometer 200, bandpass filter 270 filters frequency comb light244 to eliminate frequency components outside a certain range. Thefilter bandwidth can be from 0.001 nm to 1500 nm, more specifically from1 nm and 2 nm.

In deformometer 200, electrooptic modulator 276 can include afiber-based electrooptic crystal driven to phase modulate lighttravelling through the fiber. Phase modulation depths generated byelectrooptic modulator 276 can be from 0.01 to 0.1.

In deformometer 200, imager 280 can include a camera or charge-coupleddevice (CCD) that produces image signal 282 that can include a pictureor movie of filtered, combined light 240 to ensure that the spatial modematches the fundamental spatial mode of the cavity 212.

In deformometer 200, acoustic optical modulator 290, in combination withlens 232, mirror 234, waveplate 236, polarizing beam splitter 236, beamdump 406 m or iris 298, shifts the single frequency of an otherwisefixed frequency laser. Light 222 incident on polarizing beam splitter236 transmits through, incident upon the acousto-optic modulator, whichproduces several frequency shifted beams spatially separated and thatare spatially filtered using a mirror and beam dump or iris. A selectedfrequency shifted beam transmitted by a lens 232 that redirects itsdirection parallel to the input beam 222.1 and focuses through waveplate248 onto mirror 234, the reflection from which retransmits to waveplate248. The combined effect of which rotates polarization by 90 degrees andretransmits to the lens and the acousto-optic modulator, which imparts asecond frequency shift to the beam and directs this second frequencyshifted beam along the same axis but opposite direction of the inputbeam.

In deformometer 200, oscillator 300 produces oscillating electricalsignal 302 for electrooptic modulator 276. The electrical signalfrequency is within a range of operation of the electrooptic modulator,and the power is within the range of operation of the electroopticmodulator.

In deformometer 200, heater 310 can include a resistive heater tostabilize a temperature of cavity 212. Electrical connection to theresistive heater is made through signal coupler 312.

In deformometer 200, pump 314 can include a pump to evacuate the gasfrom the chamber. A base pressure range of pump 314 can be, e.g., from10⁻⁶ Pa to 10⁻¹ Pa.

In deformometer 200, chamber 316 and its walls 320 encloses the chamberinterior 318 such that gasses injected into 318 are confined. Moreover,chamber 316 allows for the insertion of light into the optical cavities212. Chamber 316 can be a copper box with vacuum-compatible glassviewports, which are anti-reflective coated and wedged to minimize backreflection. For optimum performance, chamber 316 can have a leak ratesless than 1 μPa L/s.

In deformometer 200, stem 350 provides structural support and mount forcavity 212. In an embodiment, it is a ULE glass tube bonded to body 212by silicate bonding.

In deformometer 200, flow channel 324 can include a channel that allowsthe flow of gas into or out of chamber 316. Exemplary flow channelsinclude stainless steel tubing, copper tubing, and the like. In anembodiment, a leak rate of flow channel 324 are not greater than a leakrate of chamber 316.

In deformometer 200, gas line 326 can include plumbing that conducts gasfrom the gas source 330 to the chamber 316. Exemplary gas lines includestainless steel tubing.

In deformometer 200, valve 328 can include a mechanism such as a valvethat stops flow of gas past it. Exemplary valves include metal sealvalves, gate valves, ball valves with rubber seals, and the like.

In deformometer 200, gas source 330 provides a source of gas 332 and canbe a gas tank, gas production unit, and the like.

In deformometer 200, reference gas 334 can include a gas at constantpressure throughout the measurement. Exemplary reference gasses includevacuum, helium, nitrogen, argon, neon, krypton, dry air, and the like.In an embodiment, reference gas 334 is air at a pressure that is, e.g.,less than 10⁻⁴ Pa.

In deformometer 200, probe signal 400 can include a Pound-Drever-Hallsignal used as feedback on the output of laser 220 to make laser light222 resonant within optical cavity 200?

In deformometer 200, reference signal 402 can include an oscillatingsignal between optical frequency comb 244 and light 222, allowing forabsolute measurement of the frequency of light 222.

In deformometer 200, mixed optical signal 404 can include an oscillatingsignal whose frequency of oscillation is determined by the differencefrequency of the two components of light 222 that compose it.

In deformometer 200, beam dump 406 can include an object that absorbslaser light such as a blackened screen, Wood's horn, and the like.

In deformometer 200, first pressure P1 and second pressure P2 are thepressures at which gas species 332.1 and 332.2 are inserted in thepressure chamber 316.

In deformometer 200, optical cell optic 410 can include an opticallytransmissive material that transmits both components of combined light240 and shifted combined light 424. Exemplary elements include sapphirewindows, Pyrex windows, and the like. Moreover, the windows can bemounted normally or at angle, with specific benefit being Brewster'sangle to minimize reflection.

In deformometer 200, cell body 414 can include mechanical object thatattaches optical cell optic 410, provides for an opening for gas line326, and holds gas inserted. The pressure contained in the cell can befrom 0 to 10 MPa, specifically from 1 kPa to 10 MPa, and morespecifically from 100 kPa to 3.6 MPa. Cell body 414 can have a leakrates that is less than 1 mPa L/s.

In deformometer 200, shifted combined light 424 is light that has beenphase-shifted by the presence of the gas in cell 414.

In deformometer 200, interfered combined light 426 is a combination ofinterfered light 428, each of which is itself a product of opticalmixing between the individual wavelength components of combined light242 and shifted combined light 424. Interfered light 428 is zero whenthe phase difference of its components in 242 and 424 have a relativephase shift of π (or an odd multiple thereof) and maximal intensity whenits components in 242 and 424 have a relative phase shift of 0 or 2π (orany even multiple thereof).

In deformometer 200, phase detector 412 is a detector the measures thephase change of shifted combined light 424 due to the insertion of gasby analyzing the signal from photodetector 238 and comparing it relativeto the maximum signal. Specifically, it takes the inverse sine functionof the intensity of the light 428 relative to the maximum possiblevalue.

Deformometer 200 can be made in various ways. In an embodiment, aprocess for making deformometer 200 includes: disposing optical cavity212 in chamber 316 that provides chamber interior 318 bounded by wall320; disposing laser 220 in optical communication with optical cavityoptic 210; optionally interposing lens 232, optical isolator 256, fibercoupler 258, propagation coupler 272, filter 270, polarizer 249,waveplate 248, mirror 234, beam splitter 236, optical combiner 230,fiber optic 274, electrooptic modulator 276, or amplitude opticalmodulator 290 between laser 220 and optical cavity optic 210; disposingcavity signal 228 in optical communication with optical cavity optic210; and optionally disposing beam splitter 236 or optical combiner 230between optical cavity optic 210 and light detector 238 or imager 280.The process can include aligning laser 220 with optical cavity 212 andlight detector 238. The process can include connecting optical cavityoptic 210 or wall 320 to a gas handling system such that gas source 330and pump 314 are disposed in communication with various elements, and apressure monitoring system can monitor pressure thereof.

Deformometer 200 has numerous advantageous and unexpected benefits anduses. In an embodiment, a process for determining deformation of opticalcavity optic 210 disposed on optical cavity 212 with deformometer 200includes combining first light 222.1 with second light 222.2; producingcombined light 240 from first light 222.1 and second light 222.2;receiving, by entry optical cavity optic 210.1 disposed at an entry end216 of a cavity body 214 of a deformometer 200, combined light 240;transmitting, from first optical cavity optic 210.1 to exit opticalcavity optic 210.2 disposed at exit end 218 of cavity body 214, combinedlight 240, entry optical cavity optic 210.1 being in opticalcommunication and optically opposing exit optical cavity optic 210.2;receiving, by exit optical cavity optic 210.2, combined light 240;producing filtered combined light 224 from combined light 240transmitted by first optical cavity optic 210.1 and second opticalcavity optic 210.2; producing, from filtered combined light 224, firstfiltered light 226.1 and second filtered light 226.2; and analyzingfirst filtered light 226.1 and second filtered light 226.2 to determinedeformation of entry optical cavity optic 210.1 and exit optical cavityoptic 210.2. Here, analyzing first filtered light 226.1 and secondfiltered light 226.2 includes determining the frequency of the firstfiltered light and second filtered light.

The process for determining deformation with the deformometer 200includes determining the frequency shift of the cavity upon insertion ofgas. Upon disposal of gas in optical cavity 212, the frequency shift ofa Fabry-Perot cavity is

$\begin{matrix}{{{n_{i}\left( {p,\lambda} \right)} - 1} = {{{f_{FSR}\left( {1 + \epsilon_{\alpha}} \right)}\frac{\frac{\left( {f_{i} - f_{f}} \right)}{f_{FSR}} + {\Delta\; m}}{f_{f}}} + {n\; d_{M}}}} & (1)\end{matrix}$wherein n_(i)(p,λ) is the index of refraction of gas species i, assumedto be known for a given pressure p and wavelength λ; f_(FSR) is the freespectral range of the cavity; ∈_(a) is the dispersion of the mirrors,d_(m) deformation due to the mirrors; f_(i) and f_(f) are the initial(vacuum) and final (pressurized) frequencies; Δm is the change in cavitymode number, and d_(M) is the deformation of the cavity due to thechange in pressure. The effective fractional frequency change follows.

$\left( \frac{\Delta\; f}{f} \right)_{eff} = {{f_{FSR}\left( {1 + \epsilon_{\alpha}} \right)}\frac{\frac{\left( {f_{i} - f_{f}} \right)}{f_{FSR}} + {\Delta\; m}}{f_{f}}}$The effective fractional frequency change simplifies Eq. 1 to

$\begin{matrix}{{{n_{i}\left( {p,\lambda} \right)} - 1} = {\left( \frac{\Delta\; f}{f} \right)_{eff} + {n\; d_{M}}}} & (2)\end{matrix}$

Deformometer 200 can include lasers that provide different wavelengthsof light to interrogate the index of refraction of a gas in opticalcavity 212. If the dispersion of the gas is known, multiple measurementsare used to determine deformation of optical cavity 212 in an absence ofany outside assumptions or calculations about the material properties ofoptical cavity 212. In an embodiment, optical cavity 212 is filled withgas; two or more lasers 220 providing light 222 at differentwavelengths. Lights 222 advantageously are well-separated in wavelength.A gas handling system provides and communicates a purified gases intooptical cavity 212, and gases can be evacuated from optical cavity 212.A pressure monitoring system monitors constancy of pressure in opticalcavity 212. This system need not measure the pressure absolutely. Anoptical frequency detector monitors optical cavities and opticalelements and measure a frequency shift thereof upon disposal of gas intooptical cavity 212.

${{n_{i}\left( {p,\lambda_{1}} \right)} - {n_{i}\left( {p,\lambda_{2}} \right)}} = {\left( \frac{\Delta\; f}{f} \right)_{{eff}\;,1} - \left( \frac{\Delta\; f}{f} \right)_{{eff}\;,2}}$where

$\left( \frac{\Delta\; f}{f} \right)_{{eff}\;,i}$is the effective frequency shift measured at wavelength λ_(i). Additionof the two effective frequencies provides deformation of optical cavity212. Subsequently, after deformation is measured the measurement ofproperties of the gas inside optical cavity 212 become primarymeasurements.

In an embodiment, with respect to FIG. 1, the frequency shift ismeasured directly using the frequency detectors 237. The frequency shiftcan be measured relative to a stable reference frequency via mixing ofthe optical signals on light detector 238. The reference can include afrequency comb, a second, independent reference cavity, the referencecavity on a fixed-length optical cavity (FLOC), a wavelength meter, andthe like.

In an embodiment, with respect to FIG. 2, replace the optical frequencydetectors 237 by light detectors 238. The light detectors are used infeedback to lasers 220 to ensure that combined light 240 equals combinedfiltered output light 224 when gas is inserted and removed from opticalcavity 212. Frequency shift of light 240 of the laser light 222 are thenmeasured using frequency comb light 242 as a reference by measuring beatfrequencies between frequency comb light 242 and laser light 222 asdetected on optical power detector 238. In this case, the frequencyreference against which all frequencies are measured is the frequencycomb 224.

In an embodiment, with reference to FIG. 3, include a HeNe laser 220.1producing laser light 222.1 at wavelength 633 nm and external cavitydiode laser 220.2. Electrooptic modulator 276 produces two frequencysidebands on light 222.2 that are reflected from cavity optic 212 and,after reflecting from beam splitter 236 are detected by photoreceiver238.5 and are used in feedback to laser 220.2 to ensure that 222.2 and226.2 are equal. Likewise, light detector 228 monitors the transmissionof light 222.1 through the cavity to ensure that 226.2 equals to 222.1.Feedback is employed on the temperature of laser 220.1 and through anacousto-optic modulator 290 to shift the frequency of laser light 222.1in order to ensure that 222.1 is equal to the resonant mode of thecavity. In this case, the frequency reference against which allfrequencies are measured is the frequency comb 224.

In an embodiment, with reference to FIG. 4, fixed-length optical cavity(FLOC) has reference cavity 212.2 interrogated by laser light 222.3 and222.4 from lasers 220.3 and 220.4, respectively. The frequency of lightfrom 220.3 and 220.4 does not deviate from 220.1 and 220.2,respectively, by more than the bandwidth of light detector 238. Theselasers have their light equal to the filtered cavity light on the outputof the reference cavity of the FLOC. These lasers then become thefrequency reference for lasers 220.1 and 220.2, and the frequency shiftsare measured with respect to these by detecting the difference frequencyvia optical mixing on light detectors 238.8 and 238.9.

Having more than two wavelengths probing the cavity is advantageous, asit provides additional information regarding the deformation. In anembodiment, with reference to FIG. 5, an optical frequency comb 244replaces the individual laser sources 222. The comb with frequencycomponent spacing considerably less than or equal the optical cavitymode spacing can be used to interrogate the cavity at vacuum and withgas. The reference signal 252 is used to control the absolute frequencyof the comb and the relative of frequency components. The comb isscanned and the transmission of the comb through the cavity is measuredusing the Fourier transform spectrometer 250. The relative spacing ofthe cavity peaks determines the cavity dispersion of the cavity when nogas is present. After injection, the relative spacing of the cavitypeaks is determined by the sum of the dispersion of the cavity, thedispersion of the gas, and the deformation. If the dispersion of the gasis known and the dispersion of the cavity is measured, the deformationis measured by subtraction of the gas and cavity dispersion.

In an embodiment, with reference to FIG. 6, a fixed-length opticalcavity (FLOC) 212 that receives two gases, e.g., He and N2, and a HeNelaser at 633 nm. The frequency reference for the FLOC was the FLOC'sreference cavity and the mode of detection for the frequency shift wasbeat-note detection. FLOC 212 is a Fabry-Perot cavity. Both gasesexperience the same deformation, by subtracting the two one cancels thedeformation and instead measures the difference between the index ofrefraction of the two gases as provided in the following equation.

${\left( {n - 1} \right)_{He} - \left( {n - 1} \right)_{N_{2}}} = {\left( \frac{\Delta\; f}{f} \right)_{{eff},{He}} - \left( \frac{\Delta\; f}{f} \right)_{{eff},N_{2}}}$providing extraction of the gas density. Likewise, adding the two

${\left( {n - 1} \right)_{He} + \left( {n - 1} \right)_{N_{2}} - {\left( {n_{He} + n_{N_{2}}} \right)d_{m}}} = {\left( \frac{\Delta\; f}{f} \right)_{{eff},{He}} + \left( \frac{\Delta\; f}{f} \right)_{{eff},N_{2}}}$provides the deformation of the gas on the optical element. As with thetwo-color method, addition of the two effective frequencies providesdeformation of optical cavity 212.

In an embodiment, with respect to FIG. 7, optical cavity 212 is replacedby optical cell 412 and the measurement becomes a phase measurement.Upon insertion of the gas into optical cell 412, the shifted lightchanges phase Δϕ according to

${{n_{i}\left( {p,\lambda} \right)} - 1} = {\frac{\left( {\Delta\;\phi} \right)\lambda}{8\;\pi\; L} - \frac{2{d_{M}(p)}}{L}}$where L is the length of optical cell 412. When the phase change Δϕ_(i)is simultaneously measured at two wavelengths i=1, 2, subtraction of thetwo yields the index of refraction and the pressure,

${{n_{i}\left( {p,\lambda_{1}} \right)} - {n_{i}\left( {p,\lambda_{2}} \right)}} = {\frac{\left( {\Delta\;\phi_{1}} \right)\lambda_{1}}{8\;\pi\; L} - \frac{\left( {\Delta\;\phi_{2}} \right)\lambda_{2}}{8\;\pi\; L}}$and addition yields the deformation through

${{n_{i}\left( {p,\lambda_{1}} \right)} + {n_{i}\left( {p,\lambda_{2}} \right)} - 2} = {\frac{\left( {\Delta\;\phi_{1}} \right)\lambda_{1}}{8\;\pi\; L} - \frac{\left( {\Delta\;\phi_{2}} \right)\lambda_{2}}{8\;\pi\; L} + \frac{4{d_{M}(p)}}{L}}$This measurement is a phase measurement, and the phase reference throughwhich changes in the phase are measured is the stable arm of theinterferometer that does not pass through the cell.

Deformometer 200 and processes disclosed herein have numerous beneficialuses, including primary pressure metrology, self-calibration of pressuretransfer standards, self-calibration of deformation effects due to otherexternal forces. Advantageously, deformometer 200 overcomes technicaldeficiencies of conventional articles such as a Fabry-Perotrefractometer or a fixed-length optical cavity, wherein a measurementaccuracy of which is limited by deformations of the optical elements bydeformation of optical elements when gas is injected.

Moreover, deformometer 200 and processes herein have numerousadvantageous properties. In an aspect, deformometer 200 can be used tomeasure refractive index and dispersion of unknown gasses. Oncedeformation of the optics due to gas pressure are known, which can bedetermined using a gas with known dispersion, deformation will remainconstant for all gasses. Thus, a gas with unknown dispersion can bemeasured sequentially, using the same deformation parameters determinedusing a known gas.

Deformometer 200 and processes herein unexpectedly eliminate calibrationof Fabry-Perot refractometers and fixed-length optical cavities aspressure standards, by removing a source of uncertainty in a pressuremeasurement. Moreover, deformometer 200 provides a potential primarystandard for pressure from 0.1 mPa and 3.6 MPa.

The articles and processes herein are illustrated further by thefollowing Examples, which are non-limiting.

EXAMPLES Example 1. Optical Cavity for Realization of the Pascal

A fixed-length optical cavity provides a relatively simple andstraightforward mechanism for determining the refractive index of a gas.If a laser is servo-locked in resonance with a Fabry-Perot cavity filledwith gas, the frequency f of the laser is given approximately by:

$\begin{matrix}{{f = \frac{m\; c_{0}}{2{nL}}},} & (33)\end{matrix}$where m (the mode order) is the integer number of wavelengths in thecavity, and L is the length of the cavity. If gas density changes,causing n to change, the servo adjusts f so as to maintain resonancewith the cavity. Hence frequency changes track changes in refractiveindex, in a manner governed by Eq. (33). If a measurement of the initialfrequency f_(i) is made at vacuum, where n=1 exactly, and a secondmeasurement f_(f) is carried out at some pressure of interest,refractivity (n−1) can be determined from by Eq. (33) applied to the twofrequency measurements:

$\begin{matrix}{{\left( {n - 1} \right) = \frac{f_{i} - f_{f} + {\Delta\;{m\left( \frac{c_{0}}{2L} \right)}}}{f_{f}}},} & (34)\end{matrix}$

where Δm is the change in mode order between the initial and finalmeasurements. A traditional method of determining the length L of thecavity or mode order m is to measure of the free spectral range (FSR) ofthe cavity; FSR measurements might be used to determine m and thus Δm,with proper consideration given to the effect of dispersion in the gas.Other approaches can also be used to determine Δm, where the simplest isto rely on approximate knowledge of the pressure, as measured by anancillary gauge of modest accuracy, to derive an estimate for Δm that isrounded to the nearest integer.

When highest possible accuracy is required, it is also necessary toconsider additional effects not captured in Eq. (33) or Eq. (34). Inparticular, determining the physical length L of the cavity via FSRmeasurements requires consideration of mirror phase shifts and, forsub-part-per-million accuracy, diffraction effects (the Gouy phaseshift). Eq. (34) can be modified to correct for an additional effect ofsignificance, namely, the length of the cavity changes in response topressure changes. The resulting modifications to Eq. (34) due to Gouyphase, mirror phase shifts, and pressure-induced length changes areconsidered.

The frequency f of the laser is determined by measuring the frequencydifference between the laser and a reference laser of known frequency(beat frequency measurement). Lowest noise and drift and the leastsensitivity to thermal variations can be achieved if the reference laseris servo locked to a second cavity, built on the same spacer as themeasurement cavity, and held at vacuum at all times.

A fixed-length optical cavity (FLOC) following the principles describedabove is described. The cavity is shown in FIG. 7. Four mirrors aresilicate bonded to the ends of a spacer. Two of the mirrors are bondedto the ends of the slot at the top of the device. This forms aFabry-Perot cavity that is open to its surroundings and contains the gasfor which the pressure is to be measured. In practice, the entire cavityis held in a copper chamber that forms a vacuum chamber. The other twomirrors are bonded to the ends of a hole to form an enclosed referencecavity that can be continuously pumped to vacuum. A pedestal (notvisible in FIG. 7) is bonded to the bottom of the spacer body with ahole for pumping the reference cavity. The spacer body, mirrorsubstrates, and pedestal are made from Corning ULE (Ultra Low Expansionglass), silicate bonded together to form a near-monolithic structure. Inequilibrium, the chamber is isothermal at the millikelvin level. Thevolume of the enclosure is minimized so as to minimize pV work that willdisturb the thermal equilibrium when gas is admitted to the chamber.

When the FLOC is used with a gas of known refractive index, theachievable uncertainty is limited by variations in the length of thecavities with changing pressure. The changes in length of themeasurement cavity are almost entirely due to the bulk modulus of thematerial. The reference cavity has additional changes due to bending ofthe windows (the mirror substrates). These effects are easily correctedthrough a combination of measurements and calculations, butuncertainties in the bulk modulus and in the position of the resonantmodes in the reference cavity limit the attainable accuracy.

The FLOC has a broad range of operation from less than one pascal to 3MPa. Virial coefficients for helium are known with sufficient accuracythat the refractivity as a function of pressure can be calculated up to3 MPa with less than 1×10⁻⁶ relative uncertainty which indicate that theFLOC could potentially replace part of the pressure scale thattraditionally has been dominated by piston gauges. The variation of bulkmodulus with pressure will cause a nonlinearity that can be considered;this has been measured for ULE. The resulting nonlinearity in pressuremeasurement is about 10 μPa/Pa at 3 MPa. The uncertainty of thiscorrection is currently estimated to be less than 1 μPa/Pa.

On the low-pressure side, the FLOC has an operating range extending downto the regime of ionization gauges. When compared to an ionizationgauge, the FLOC noise level was only 0.1 mPa for 1 s averages. Inprinciple, sub micropascal measurements can occur. Lasers locked toFabry-Perot cavities near room temperature have achieved fractionalfrequency stabilities below 10⁻¹⁶ for averaging times on the order of 1s, where the basic limitation is thermal noise (in the mirror substrate,the coatings, and the cavity spacer). For typical gas species such asnitrogen, oxygen, and water vapor, this thermal noise floor correspondsto about 4×10⁻⁷ Pa. However, the stability degrades rapidly as themeasurement time increases, and as a practical matter the measurementfloor will be limited by thermal or temporal instabilities of the cavityand by outgassing.

The FLOC is a device that can be subject to pressure-induced changes inthe mechanical length of the cavities, which can be measured andcorrected or calculated from known values of material elastic constants.For pressures above 100 Pa, best performance can be achieved if thepressure distortions are evaluated by comparison of FLOC pressuremeasurements to the VLOC.

FLOC can be an independent standard using two gasses for which therelationship of refractivity to pressure is well known, either fromtheory (for helium) or from experimental measurements (potentiallymeasured to high accuracy with the VLOC). This can be done withoutrequiring direct comparison to a known pressure standard, working underthe assumption that the only significant error in the FLOC is pressuredistortion. If, for example, measurements of refractivity are made forhelium and nitrogen with both at the same pressure (where the pressureneed not be known), then the two measured refractivities can be used todetermine two unknowns; (a) the unknown pressure and (b) the pressuredistortion of the FLOC. In effect, the known refractivities of the twogasses serve as a mechanism for traceable dissemination of the Pascalwhere traceability is provided by the atomic/molecular properties of thegasses in place of a direct comparison to a primary standard. (Except inthe case of helium, there is indirect comparison to the pressurestandard that was used when the gas refractive index was determinedexperimentally.) The uncertainty of pressure measurement using thetwo-gas dissemination method is potentially a few parts in 10⁶ atatmospheric pressure, if the refractivity of nitrogen can be measuredwith comparable uncertainty.

FLOC can be tested with three known gasses to uncover contamination ofone of the gasses used for pressure dissemination. Three-gas testscombined with other internal consistency checks can be used to verifycontinued proper operation of all components of the system other thanthe temperature sensor. With this distortion correction, the FLOC shouldbe able to extend the high accuracy of the VLOC to a much broaderpressure range, conceivably spanning 9 orders of magnitude frommillipascal to megapascal.

Example 2. Quantum-Based Vacuum Metrology

In this example, equations are numbered starting at Eq. 1.

Traditionally pressure is defined as a force per unit area, but aspressures extend further and further below an atmosphere (deeper intothe vacuum) this definition becomes increasingly inconvenient andimpractical. Instead, at low pressures the pascal is realized throughthe ideal gas law,p=ρ _(N) k _(B) T=ρ _(V) RT  (1)where ρ_(N) is the number density of particles and ρ_(V) is the molardensity, R is the gas constant, and T is the temperature. In thisformulation, pressure metrology becomes a counting problem,specifically, counting particles in the vacuum by any availabletechnique. This reflects the applications as well: in the high-vacuumand below, most users are concerned with the amount of gas in thevacuum, e.g. as a contaminant, rather than the force it produces. Eq.(1) fundamentally relates pressure to the Boltzmann constant k_(B). SIunits are tied to defined physical constants, e.g. Plank's constant orthe speed of light in vacuum. Furthermore, there is an accompanyingshift away from electronic to photonic measurements. Measuring photonsinstead of electrons has several inherent benefits: optical signals aregenerally less to prone to pick-up noise from stray signals than areelectrical signals, especially for long transmission distances. Photonicsignals are high-fidelity, and can travel farther without regeneration.Additionally, optical fiber is lighter and has a larger bandwidth percross-sectional area than copper wire, and can better handle harshconditions, and so it has practical advantage, especially for use inaircraft or launch vehicles. Photonic measurements can be readilymultiplexed and allow remote interrogation. Furthermore, photons can beused to directly probe the electronic states of atoms or molecules, andto prepare quantum states, making them the tool of choice forfundamental quantum measurements.

At pressures from about at atmosphere to the high vacuum, classicalmetrology technologies are mature and can deliver uncertainties at thelevel of a few parts in 10⁶, generally adequate for stakeholders. Anadvantage of pressure standard based on the FLOC technique is that ithas the perspective to replace traditional mercury manometers, which areoften used in the vacuum range of 10⁻³ Pa to 10⁵ Pa, thus removing toxicmercury from the calibration lab. The primary high-accuracy manometersused in this pressure range also tend to be rather large, expensive, andrequire a high level of expertise to operate, and are thus usually ownedand operated by national metrology institutes or sophisticatedcalibration laboratories. The FLOC and the other quantum-SI techniquescan be a portable primary standard.

For measurements at atmospheric pressures and into the low vacuum,manometry is the traditional technique. Manometers operate on theprinciple that a fluid in a column sealed at the top will create avacuum in the sealed end of the column when it experiences the downwardforce due to its own weight. The pressure on the other end of the column(the pressure of interest, often atmosphere) exerts a force that mustbalance the gravitational force, for the fluid to be in equilibrium. Thepressure in pascal is then P=ρ_(f)gh where ρ_(f) is the fluid density, gis the local acceleration due to gravity, and h is the column height.The determination of column height is done using an ultrasonictechnique, and care is taken to minimize uncertainty from other sourcesincluding temperature. These instruments can claim relative standarduncertainties as low as 3×10⁻⁶ as demonstrated in an international keycomparison.

Several laser-based interferometer techniques are under study tointerrogate the refractivity n−1 of a gas (n is index of refraction)which is a proxy for the gas density ρ_(N), and ultimately the pressurep through the equation of state:p=k _(B) Tρ _(N)(1+B _(ρ)ρ_(N) +C _(p)ρ_(N) ²+ . . . )  (5)where k_(B) is the Boltzmann constant, T is thermodynamic temperature,and the deviations from the ideal gas law arising from two- andthree-body interactions are considered by density virial coefficientsB_(ρ) and C_(ρ). For helium gas, the virial coefficients in (5) arecalculable through statistical mechanics at a level that contributesless than one part in 2×10⁷ to the uncertainty of pressures below 1MPa.⁵⁰ Current state-of-the-art thermodynamic thermometry implies thatthe thermal energy k_(B)T can be measured better than one part in 10⁶.Therefore, with the highest accuracy measurements of heliumrefractivity, uncertainties from theory and thermodynamic temperatureimply that the pascal can be realized with uncertainty at the one partin 10⁶ level, which would place it competitive with state-of-the-artpiston gauges at 1 MPa, and better than state-of-the-art mercurymanometers at 100 kPa and below.

Depending on the details of these approaches, the techniques describedherein result in a device that is considered alternatelyfunctionally-primary, primary, or a transfer standard. In all cases, twomajor obstacles must be overcome which are discussed below: Thepressure-dependent index of refraction must be known to high accuracy,and any distortions in the measurement device must be accounted for. Webegin with a brief discussion of the underlying physics before turningto a description of several experimental devices. The speed of lightwith frequency ν in a gas, c, is reduced from that in an ideal vacuum c₀by a coefficient n, that is,c=c ₀ /n.  (6)

The mechanism by which this happens concerns the polarizability of theparticles constituting the gas. Such polarizabilities are the quantumbasis of the method, and our ability to calculate the polarizability ofhelium and thus its refractivity is ultimately what makes the techniquedescribed herein a fundamental standard, consistent with the quantum-SI.Theoretic determinations of these fundamental atomic properties wereperformed at relativistic and quantum electrodynamics (QED) levels.Extending the method to gases other than helium is done in a ratiometricway that preserves the fundamental nature of the method.

The relation of n to ρ_(N) for an isotropic homogeneous medium isobtained by the Lorentz-Lorenz equation,

$\begin{matrix}{{\frac{n^{2} - 1}{n^{2} + 2} = {{\frac{1}{3\; ɛ_{0}}\rho_{N}\alpha} = {A_{R}\rho_{V}}}},} & (7)\end{matrix}$where α is the dynamic polarizability of an individual molecule of gasin the volume, A_(R) is a virial coefficient, the molar dynamicpolarizability, and ε₀ is a fixed physical constant, the vacuumdielectric permittivity. By determining index refraction, we can realizeρ_(V). To calculate polarizability from first principles requiresconsidering relativistic, QED, and finite mass effects, and this hasbeen done for both the polarizability and refractive index of helium toan uncertainty of below one part in 10⁶. (note that for accuracy on theorder of one part in 10⁶, it is also necessary to include the effect ofmagnetic susceptibility, which is omitted in Eq. (7) for simplicity.

Pressure sensors based on refractometry can in principle be based on anygas and He has the advantage that it's pressure dependent index ofrefraction has been calculated to high accuracy, making such a deviceintrinsically absolute. However, in a practical device made of ultra-lowexpansion (ULE) glass, helium has the disadvantage that it is absorbedinto the glass. And so, a refractometer using gases other than helium,such as N₂, may be a more useful method of pascal dissemination, butfirst the index of refraction of that measurement gas must bedetermined.

Refractometers are pressure standards. The concept of index ofrefraction is that a photon with a fixed wavelength will have adifferent frequency in the presence of gas than in a vacuum. A laser canpropagate in each of two channels, one filled with gas and the otherevacuated, and measures the frequency change, as performed in FLOC. Moreprecisely, a laser is wavelength-locked in resonance to a Fabry-Perotcavity, if gas density (i.e., pressure) changes, the servo adjusts thefrequency f to maintain resonance with the cavity. Changes in f thengive the index of refraction according to:

$\begin{matrix}{{{n - 1} \approx \frac{{{- \Delta}\; f} + {\Delta\;{m\left( {{c_{0}/2}L} \right)}}}{f}},} & (8)\end{matrix}$where Δf=f−f₀ (f_(o) is the laser frequency in vacuum, and f is thefrequency in the gas medium,) Δm is the change in mode order, and L isthe length of the cavity. In practice, the laser frequency in eqn. (8)is never measured directly but is determined by measuring the differencein frequency between the measurement laser and a reference laser lockedto the vacuum channel. Both the reference and vacuum channel deformunder pressure. Much of the deformation is an overall compression due tofinite bulk modulus, which is common to both the reference andmeasurement channels so that the effect largely cancels out. Anotherimportant effect is bending of the mirror surfaces in the referencechannel due to the pressure differential across these mirrors. Themeasurement equation for pressure determined by the FLOC is then:

$\begin{matrix}{{p = {\frac{1}{{\left( \frac{3}{2k_{B}T} \right)A_{R}} - d_{m} - d_{r}}\left( \frac{f_{vac} - f_{gas}}{f_{gas}} \right)}},} & (9)\end{matrix}$where f_(vac) (f_(gas)) is the frequency in the evacuated (gas-filled)cavity. The distortion term d_(r) is essentially the fractional changein length of the reference cavity when gas is added to the cavity (anegative number). Similarly, d_(m) is the negative of the fractionalchange in the measurement cavity length (a positive number, where thesign is an artifact of the derivation). For simplicity, Eq. (9) onlyretained terms of order Δf/f. The correction for the distortion termsare approximately d_(m)≈d_(r)≈1.1×10⁻¹¹ Pa⁻¹, whereas the index n varieswith p by 3.2×10⁻¹⁰ Pa⁻¹ for helium at 303 K. Note that the twocorrection factors cancel each other within 10%. Therefore, without anycorrection for the distortion, the FLOC is a primary standard forpressure to about 0.3%.

FIG. 9 shows (a) dual FP cavity refractometer in its thermal/vacuumapparatus: the pressure measurement cavity is in gas, and the referencecavity is ion-pumped to high-vacuum, and panel (b) is a photograph ofthe refractometer. Panel (c) shows distortions in cavity lengths perpascal of pressure on the measurement cavity when the reference cavityis at vacuum.

Performance can be achieved by measuring two or more different gases ofknown refractivity at a certain pressure. Both the cavity distortion andthe absolute pressure can be determined, since measurements of twogasses provide two equations in the two unknowns. Helium refractivity isknown as a function of pressure by calculation. When a measurement ismade using two gasses, the FLOC provides traceability to primary methodsand becomes functionally primary in the important sense that it neverneeds to be calibrated against a pressure standard. Thus, the invariantatomic/molecular properties of the gasses (i.e., refractivity) willserve as a practical functional standard for universal dissemination ofthe Pascal. In past work, the FLOC demonstrated ((2mPa)²+(8.8×10⁻⁶p)²)^(1/2) expanded uncertainty as a transfer of thepascal, and so the FLOC as a transfer standard of the pascal outperformsthe manometer at pressure below about 1 kPa. FIG. 10 shows adisagreement in pressure as measured by two separate laserrefractometers (pFP) and mercury ultrasonic manometer (pUIM). The dashedlines are the manometer uncertainty.

FLOC is a primary pressure standard when used with helium gas, butdistortion of the optical cavity and mirrors, including dynamic effectscaused by diffusion of helium into the ULE glass include uncertainty.Even if the measurement gas is nitrogen or some other species that doesnot diffuse into the glass, distortion is accounted for. A refractometercan be a primary standard with performance of characterization of thedistortion.

With reference to FIG. 11, an optical technique finds the laser beamlocation on the mirror surface, and shape is calculated through afinite-element analysis. From this, a bending profile is extracted. Bycombining the bending profile with knowledge of the beam location, anestimate can be made of the distortion error in the FLOC. Here,correcting FLOC distortion is performed by finite-element analysis andan inspection of the mode position on the mirror. Panel (a) shows animage of the mirror with a bond interface. Through edge-detection, anestimate can be made of the area upon which the pressure acts. In panel(b), another image is shown with a laser beam aligned to the cavityresonance. By combining these two images, an estimate of the location ofthe beam on the mirror surface is made. The result of a finite-elementanalysis is shown in panel (c), wherein datasheet values were used forelastic properties of ULE glass, and the geometry was estimated by thebond line in panel (a). The difference in mirror bending calculated byfinite-element is extracted as a profile, as shown in panel (d).

It is contemplated that elastic properties of the glass can bedetermined directly by mechanical means, using resonant ultrasoundspectroscopy. Achieving relative uncertainty lower than one part in 10⁵in helium refractivity would require determination of the bulk moduluswithin 0.03%. Additionally, doping inhomogeneities in ULE (i.e., givingrise to variations in the coefficient of thermal expansion) involves atoken whose elastic properties are measured by mechanical means that caninclude a reflection of the elastic properties of the FP cavity itself.

Multi-wavelength interferometry and calculated dispersion of helium candetermine FLOC deformation. This can be accomplished by interrogatingthe FLOC with two laser frequencies ν₁ and ν₂ locked to the opticalcavity, which has the advantage that it can be done in-situ. Themeasurement equation for pressure determined by the FLOC under theseconditions to first order is:

$\begin{matrix}{p = {\frac{1}{\delta\;\alpha}{{\frac{k_{B}T}{2\;\pi}\left\lbrack {\left( \frac{\delta\; v_{1}}{v_{1}} \right) - \left( \frac{\delta\; v_{2}}{v_{2}} \right)} \right\rbrack}.}}} & (10)\end{matrix}$

Here δα is the change in the atomic polarizability between the two laserfrequencies at the same gas pressure p. Again, the atomic polarizabilityα(λ), where λ=c/ν, is known for He from fundamental theoreticalcalculations. In Eq. (10), deformation terms that were present in Eq.(9) have cancelled and thus, using two lasers, we now have a primaryFLOC. The two-laser method involves dispersion that is a small effectcompared to refractivity. For two practical laser frequencies, say 633nm (HeNe laser) and 1550 nm (standard telecom wavelength), thedifference in n−1 is approximately 1.6×10⁻⁷ (at atmospheric pressure),which is more than 200 times smaller than the value of n−1. Some sourcesof noise and systematic uncertainties will increase, and the currentstate of theory and calculation of helium dispersion would limit theapproach to 5 parts in 10⁶. A primary FLOC can include this multi-colortechnique.

Deformations of FLOC can be avoided or corrected in refractometers ofalternate design. One such design is the Monolithic Interferometer forREfractometry (MIRE). A feature of the apparatus is threeinterchangeable triple-cells of different length as shown in FIG. 12(a),but almost identical geometries, material properties, and location ofthe laser beams through all windows. This feature is designed to makethe window distortion common-mode in measurements of helium refractivityperformed in cells of different lengths, and allowed cancellation oferror to 1.8%, which resulted in a 9.8 ppm relative uncertainty in therefractometer. When the uncertainty in the refractometer was combinedwith the uncertainties in the thermodynamic temperature of helium, gaspurity, and the Boltzmann constant, the total standard uncertainty inthis primary realization of the pascal was 11.7 ppm. With reference toFIG. 12, panel (a) shows MIRE apparatus, and panel (b) showsrefractometry cells of three different lengths but which are otherwisenominally identical. Each borehole has a gas inlet and outlet.

Example 3. Refractometry Using a Helium Standard

In this example, equations are numbered starting at Eq. 1.

Measuring air refractive index can provide realization of the meter viainterferometry. Laser wavelengths in vacuum can be measured to highaccuracy, but the wavelength in air, which serves as the basic metricfor almost all interferometry, requires knowledge of refractive index.Most commonly, refractive index is determined by measuring air pressure,temperature, humidity, and possibly carbon dioxide concentration, andcalculating the refractive index using Edlen or similar equations. Withsome care we can routinely determine refractive index in a uniformlaboratory environment to 1 part in 10⁷, and with state-of-the-artenvironmental sensors the expanded (k=2) uncertainty might be reduced aslow as 2 parts in 10⁸. If yet smaller uncertainty is needed, severalalternate approaches could be pursued. For example, it may be possibleto carry out the measurements in vacuum, or to perform measurements in ahelium environment. Helium has two advantages relative to air: (1) therefractive index is known to very high accuracy from ab initiocalculations[2] and (2) the molar refractivity of helium is about oneeight that of air, so that for a given uncertainty of pressure andtemperature measurement, the refractive index of a helium sample can bedetermined with one eighth the uncertainty of air refractive index.

When measurement in vacuum or a helium atmosphere is not practical, afinal possible route to high-accuracy displacement measurements is toemploy a gas refractometer to determine the air refractive index. Arefractometer based on a laser locked to the transmission maximum of aFabry-Perot interferometer (hereafter designated FPI) of nominally fixedlength is described. Changes in laser frequency track changes inrefractive index as the interior of the FPI cavity is filled with gas orpumped out to vacuum. For an ideal cavity, the change in laser frequencyis proportional to the change in refractivity going from the evacuatedto the filled state. (Refractivity is n−1, where n is the refractiveindex.) Since the evacuated state has a known refractive index n=1,measuring the change in refractive index also tells us the absoluterefractive index of the gas. A practical problem in implementing thisscheme is the difficulty in accounting for distortions of cavitydimensions caused by the changing pressure. The method herein overcomesthe problem by using helium gas at atmospheric pressure as a knownreference, because the refractive index of helium can be accuratelycalculated from first principles. Since the refractive index of heliumis known, we can predict the change in laser frequency when the cavityis filled from vacuum to some helium pressure. Any deviation from thisprediction provides a measure of cavity distortion, allowing us to“error map” the refractometer as a function of pressure. Accuratelymeasure helium pressure and temperature is involved to carry out thiserror correction procedure, but the low refractive index of helium putsminimal demands on the required accuracy of the sensors. Also, thehigh-accuracy sensors do not need to be a permanent part of theapparatus, since the error is not expected to change with time.

The refractivity of helium, n−1, is very nearly proportional to themolar density p of the gas sample and to the molar polarizability ofhelium A_(R), that is, n−1 ∝ρA_(R). The proportionality is not exact indense gasses. Departures from linearity are considered in theLorentz-Lorenz equation:

$\begin{matrix}{\frac{\left( {n^{2} - 1} \right)}{\left( {n^{2} + 2} \right)} = {{A_{R}\rho} + {B_{R}\rho^{2}} + \ldots}} & (1)\end{matrix}$where BR, the second refractivity virial coefficient, accounts fortwo-body interactions. Equation (1) allows calculation the refractiveindex if the molar density is known. The molar density as a function ofpressure and temperature can be determined from the expressionsρ=P/(ZN _(A) kT)  (2)Z=1+B(T)ρ+C(T)ρ²+ . . .  (3)where P is pressure, T is absolute temperature, k is the Boltzmannconstant, N_(A) is the Avogadro constant, Z is the compressibilityfactor for a non-ideal gas, and B(T) and C(T) are virial coefficientsfor the compressibility expansion. All of the parameters to make thiscalculation are known. In particular, A_(R) is known from ab initiocalculations of the atomic polarizability. It is possible to combinecalculations of the static polarizability with dynamic polarizability todetermine a value whose uncertainty is limited by lack of QEDcorrections to the dynamic terms. As a function of wavelength λ, themolar polarizability is then given by

$\begin{matrix}{A_{R} = {0.51725407 + \frac{1197.5410}{\lambda^{2}} + \frac{3.290677 \times 10^{6}}{\lambda^{4}} + \frac{9.800874 \times 10^{9}}{\lambda^{6}}}} & (4)\end{matrix}$where A_(R) is expressed in units of cm³/mol. The estimated relativeuncertainty of this expression is 1×10⁻⁶ at optical frequencies. AtX=633 nm, equation (4) yields A_(R)=0.5202634(5) cm³/mol.

B_(R) has an effect on calculation and is known. We estimateB _(R)=−0.032-0.0001T  (5)where B_(R) is expressed in units of cm⁶/mol², the temperature T isbetween 273 K and 323 K, and the wavelength is in the vicinity of 633nm. The virial coefficient B(T) is known. Over the range 275K to 325K,the result isB(T)=13.028-0.0041T  (6)where B(T) is expressed in cm³/mol. C(T) is sufficiently small that ithas no noticeable effect on the calculations.

This provides everything to calculate the refractive index. Thecalculated refractive index for 633.0 nm radiation, at 101 325 Papressure and 20° C., is 1.000 032 426 00(8), where the standarduncertainty (8×10⁻¹¹) arises primarily from uncertainty in the Boltzmannconstant.

At 632.991 nm wavelength (the approximate vacuum wavelength of a heliumneon laser), for pressures less than 110 kPa and temperatures between273 K and 323 K, we find empirically that a good approximation to thecalculation described above isn=1+{9.38598×10⁻⁸(P/T)−1.333×10⁻¹³(P/T)²}×{0.999957+1.5×10⁻⁷ T}  (7)where P is in Pascal and T is in Kelvin. The error in this approximationrelative to the more exact procedure is nearly negligible (less than4×10⁻¹¹) over the stated range.

The Fabry-Perot refractometer is shown in FIG. 13. A laser shinesthrough a high-finesse Fabry-Perot cavity. The cavity, as shown in FIG.14, is made from a ZERODUR rod that has a channel sawed through thecenter and mirrors optically contacted to the ends. The open side of thecavity assures that the gas inside the cavity is in good equilibriumwith gas in the surrounding area. Thus, if it were placed right next tothe measurement path in an interferometer, we could expect that therefractive index inside the refractometer would closely match therefractive index in the measurement path. The mirror radii are largerelative to the length of the cavity but sufficiently small so that theyrequire only minimal parallelism of the polished ends of the rod. Thecavity is consequently easy to construct.

A laser is locked to a transmission maximum of the cavity using a simpledither of laser frequency, phase sensitive detection of the transmissionmaximum with a lock-in amplifier, and an integrating feedback loop tocontrol laser frequency and wavelength. In this manner the laser remainslocked to the center of the transmission maximum, maintaining constantwavelength even as the refractive index within the cavity changes (underthe assumption that the cavity length remains constant). To maintainconstant wavelength with changing refractive index, the servo systemmust readjust the laser frequency, and the change in laser frequency isproportional to the change in refractivity (n−1).

To determine the frequency of the wavelength-stabilized laser, theoutput is mixed with the output of a second, fixed-frequency laser; theresulting beat frequency, measured with a frequency counter, is thedifference between the unknown frequency of the wavelength-stabilizedlaser and the known fixed frequency of the second laser. An iodinestabilized laser is the fixed-frequency source.

Two independent refractometers are housed in a common environmentalchamber that can be pumped to vacuum. A gas laser with PZT lengthcontrol is locked to a 453-mm long FPI cavity and a tunable diode laseris locked to a 94 mm cavity, about ⅕ the length of the long cavity.

The basic resonance condition for a Fabry-Perot cavity is that theround-trip phase shift be an integral multiple of 2π. This phase shiftconsists of several parts. In first approximation the shift is 2π(2 L/λ)where 2 L is the round-trip length through the cavity. To this must beadded phase shifts φ_(m1) and φ_(m2) for reflection at the two endmirrors and a phase shift φ_(G) that occurs because the light within thecavity is not a plane wave. This is the Guoy phase shift with anappropriate sign convention. The resonance condition is2πm=(4πL/λ)+φ_(m1)+φ_(m2)+φ_(G)  (8)where m is some integer. The three phase shifts φ_(m1), φ_(m2), andφ_(G) represent only a very small part of the total phase shift and arenearly constant. If they are ignored, and if we make the substitutionλ=c/(nf) where f is the laser frequency, c is the speed of light invacuum, and n is refractive index, then the resonance condition takes onthe familiar formf=mc/(2nl)  (9)

When a gas is admitted to the cavity and the frequency of the lockedlaser changes from some initial value f_(i) to the final value f_(f),equation (9) implies that the final refractive index is given by

$\begin{matrix}{{n - 1} = \frac{f_{i} - f_{f} + {\Delta\;{m\left( {{c/2}L} \right)}}}{f_{f}}} & (10)\end{matrix}$

Here Δm is the change in order number, which can be determined by usingtwo cavities in a Vernier-type arrangement. Alternately, Δm might bedetermined by measuring pressure and temperature in the cavity anddetermining n with modest accuracy, sufficient to resolve any ambiguityin Δm, and then using equation (10) to improve the accuracy of n.

F is the numerator in equation (10),F=f _(i) −f _(f) +Δm(c/2L)  (11)F can be thought of as the total change in frequency of the laser, wherethe term Δm(c/2 L) includes the effect of mode hops. Note that f_(i)differs from f_(f) by no more than the free spectral range (fsr) of thecavity, because beyond this point the laser will mode hop, locking tothe next order m. Therefore, the term f_(i)-f_(f) contributes much lessto F than does the term Δm(c/2 L), where Δm may be as large as 400. Itis usually assumed that fsr=c/2 L, and thus measurement of the freespectral range provides the needed information to evaluate F. However,the dependence of the mirror phase shifts φ_(m1) and φ_(m2) on frequencycomplicates matters. The dependence of phase shift on frequency has verylittle bearing on equation (10), but the shift in φ_(m) has asignificant influence on the measurement of free spectral range. The fsris the difference in frequency of two lasers locked to adjacentlongitudinal modes of the cavity. The difference in phase shift betweenthe two modes causes a small error in determination of c/2 L which ismultiplied by a fairly large number (Δm) in equation (10). The resultingerror in refractive index is small (less than 10⁻⁹ for our 453 mm longcavity) but not negligible in some applications.

Equation (10) includes a length of the cavity that is constant, whereasthe length changes in response to pressure.

If the cavity changes length from l_(i) to l_(f), then the equation mustbe modified to read

$\begin{matrix}{{n - 1} = {\frac{f_{i} - f_{f} + {\Delta\;{m({fsr})}}}{f_{f}} + {n\left( \frac{l_{i} - l_{f}}{li} \right)}}} & (12)\end{matrix}$

The expected change in length is on the order of Δl/l=6×10⁻⁷, largeenough that it is necessary to carefully account for the effect. Thecompression is a function of the bulk modulus B of the material and ofthe change in pressure ΔP:Δl/l=(⅓)BΔP  (13)

The bulk modulus depends on Young's modulus (Y) and Poisson's ratio (ν).It can also be written as a function of Y and of the shear modulus μ:

$\begin{matrix}{B = {\frac{Y}{3\left( {1 - {2v}} \right)} = \frac{Y}{3\left( {3 - {Y/\mu}} \right)}}} & (14)\end{matrix}$

The bulk modulus is determined from measurements of shear modulus andYoung's modulus. Assuming that μ and Y are uncertain by 1%, theseuncertainties will both contribute uncertainties of about 5% in B.Combining the two uncertainties using root-sum-squares, the uncertaintyin B could then be as large as 7%, which will give rise to anunacceptably large uncertainty of 4×10⁻⁸ in the refractive index.Furthermore, for a cavity such as ours, where the mirrors have fusedsilica substrates and the body of the resonator is a polymer,commercially available as ZERODUR, the dissimilar materials causecomplicated pressure-dependent changes that are not easily predictedfrom first principles.

Avoid this problem by calibrating with helium. Since the refractiveindex of helium is known, equation (12) can be used to calculate thechange in length of the cavity (more precisely, to determine anypossible pressure-induced errors in the system), and this knowledge canbe used to correct subsequent measurements. Helium should have goodpurity, such as having less than 1 part in 10⁶ contaminants. Thecontaminants most likely to be present will shift the refractive indexby several parts in 10¹⁰.

We used He gas to measure the distortions Δl/l as a function ofpressure. Results for our long cavity are shown in FIG. 15. The dashedline shows a prediction based on the manufacturer's published values ofthe Young's modulus (90.3 GPa) and Poisson's ratio (ν=0.243). Thedeviation of experiment from the prediction amounts to an error of2.8×10⁻⁸ at atmospheric pressure. However, this deviation does notnecessarily arise from imprecise values of Y and ν. One might imaginethat additional distortions occur at the ends of the cavity as aconsequence of several possible effects. This is particularly true forour cavity because of the dissimilar materials (ZERODUR and fusedsilica) used in construction, and in fact we have seen some evidencesuggesting that the dissimilar materials are a source of distortion.Such effects will be more important in a short cavity than in a longone, and consequently a good way to check for end-effects is to compareresults from two cavities of greatly different length. Note thatequation (9) implies that fractional changes in frequency df/f are givenbydf/f=−dn/n−dl/l  (15)

In the absence of end distortion, d/l should be independent of cavitylength. Thus, if two cavities of different length share the sameenvironment (the same do/n), and in the approximation that the laserslocked to the two cavities have nearly the same frequency f the changein frequency df should be essentially identical for the two cavities.For larger changes in frequency, with df replaced by ΔF (that is, thefrequency change is corrected for mode hops) it is again true that thechange ΔF for a laser locked to a short cavity should be essentially thesame as ΔF for a laser locked to a long cavity. If we look atF_(Short)-F_(Long), the difference frequency between the lasers lockedto our long and short cavities (the intercavity beat frequency), wewould expect that the beat frequency should remain constant as helium isadmitted to the cavity. Any variations in the beat (after correction formode hops) is a very sensitive test of end-effects.

FIG. 16 shows the result of this test using our two cavities. There is avariation of about 14 MHz in the beat frequency as the cavity is filledwith helium to atmospheric pressure. 14 MHz represents a discrepancy of3 parts in 10⁸ of the laser frequency. Attribute this discrepancy to enddistortions. These distortions are probably very similar in our long andshort cavities, which are identical except for length. However,identical distortions will have about 5 times greater effect on thefrequency of our short cavity than on the frequency of the long cavity.Therefore, most of the 14 MHz discrepancy might be attributed to endeffects in the short cavity, and end effects in the long cavity areabout ⅕ as large. More precisely, under the assumption that bothcavities have the same end effects, then the 14 MHz discrepancy arisesfrom a 17.5 MHz effect in the short cavity and 3.5 MHz effect (i.e.,17.5 MHz/5) in the long cavity. Expressed as a fractional frequencyshift, the 3.5 MHz error is 7×10⁻⁹. End effect thus account for arelatively small portion of the discrepancy shown in FIG. 3, with thepredominant part of the discrepancy presumably arising fromuncertainties in the material properties.

If changes in the FPI are small (or at least predictable) over a periodof several years, then the cavity could be operated for long periods oftime without the need to evacuate the system and re-establish the vacuumfrequency. Influences that must be considered are temperature, humidity,and aging of the cavity.

With regard to aging of the cavity, the long cavity shrank at a rateΔl/l=−3.5×10⁻⁸ per year. The cavity was usually at temperatures between22° C. and 25° C. during this time, although it was briefly cycled totemperatures as high as 32° C. or as low as 14° C. The shrinkingprobably represents instability of the ZERODUR, although part of thetemporal changes might also be due to aging of the mirror coatings. TheFPI has been under vacuum for the majority of a year but was atatmospheric pressure for significant periods of time as well. We shouldnote that the ZERODUR was manufactured 6 years ago and the mirrors wereoptically contacted several years ago. ZERODUR that is younger may ageat a greater rate. In some cases, even old samples of ZERODUR have beenobserved to age at a rate of Δl/l=1×10⁻⁷ per year. Note that otherlow-expansion materials might have significantly lower rate of aging.

With regard to Thermal effects, we measure the thermal expansion of ourcavity in vacuum, as shown in FIG. 17. From frequency changes we inferthe fractional change in cavity length Δl/l, where Δl is the change inlength from the value at 20° C. The expansion coefficient indicated bythe slope of the line is 4.9×10⁻⁸/° C. About 20% of the expansion isprobably associated with end effects, as can be determined by comparingshort and long cavities in a manner analogous to what was describedpreviously for pressure variations.

The temperature graph consists of two distinct data sets. First data wastaken while the temperature was decreased slowly (over a period ofseveral days) from 26.6° C. to 24.56° C. The temperature was thendecreased rapidly to 20° C. for one day, and the remainder of the datawas obtained over a period of several days while the temperatureincreased from 20° C. to 24.47° C.

Over this temperature range the data is nearly linear, but there arenoticeable deviations from the best-fit line between 24° C. and 25° C.In this region, the data obtained during cooling lies above the linewhile the data obtained while heating lies below the line. Thedeviations, as large as 1.4×10⁻⁸, might be suggestive of eitherhysteresis or the presence of slow relaxation processes which requirelong periods of time to reach equilibrium.

We carried out an additional experiment to look more carefully atpossible relaxation and hysteresis behavior when the FPI is subject togreater temperature excursions. FIG. 18 shows the relaxation of the longcavity after heating to 30.5° C. for 48 h and then cooling to 24° C. Thegraph shows the beat frequency between the fixed-wavelength laser andthe fixed-frequency laser. The data shown begins 12 h after initiatingcooling. At this point in time the ZERODUR temperature was still about0.1° C. above its equilibrium temperature, but the data has beencorrected to constant temperature using the previously measuredexpansion coefficient. The change in beat frequency of 10 MHz seen inthe graph corresponds to a fractional change in length of 2×10⁻⁸. Thethin continuous line is an exponential fit to the data with a timeconstant of 28 h. This time constant is somewhat too slow to account forthe early data but somewhat too fast to account for later data. Theexponential fit is approaching a value of 42.9 MHz. When we cooled theFPI to 16° C. for two days and then brought it back to 24° C., the beatfrequency reached a near-constant value of 44.8 MHz after just 40 h ofwarming. In this case the approach to equilibrium is significantlyfaster than seen in FIG. 18. In general, this heating/cooling behaviordoes not seem to be completely reproducible, and at present it is notclear if the residual 2 MHz difference seen between heating and cooling(corresponding to a change of length of Δl/l≈4×10⁻⁹) should beattributed to a very slow relaxation or to permanent hysteresis. In anyevent, to assure an uncertainty better than 2×10⁻⁸ when using cavitiesof the present design, it would be advisable to wait several daysfollowing a large temperature excursion.

With regard to humidity variations, adsorption or absorption of watervapor by the mirror coatings are of potential significance when the FPIis used to measure the refractive index of moist air. We find thathumidity has an effect on the apparent length of the cavity, withvariations occurring on a time scale of days. We have studied the effectby measuring the beat frequency between the long and short cavity (as wedid for pressure) when water vapor is admitted to the chamber.Variations in the beat will occur because humidity, like other endeffects, is expected to have about 5 times greater effect on the shortcavity than on the long one. We have kept the chamber filled with watervapor for periods up to 1 week to provide time for possible slowpenetration of the coatings by vapor. In performing this test, we haveused vapor pressures as high as 3 kPa, above the saturation vaporpressure at 20° C. FIG. 19 shows variations in the intercavity beatfrequency and water vapor pressure over a period of 40 days. The graphbegins at a time when the cavity had been dried under vacuum for 1 week.The beat frequency and humidity are correlated although the correlationis obscured by time constants for changes to take place. The beatfrequency under dry conditions (at the start of the graph) decreasesrapidly on day 3 when water vapor is introduced at vapor pressures of2500 Pa to 3000 Pa. When the vapor pressure is reduced again to 2300 Paon day 4, the beat frequency slowly increases as the FPI cavities dry.When the chamber is evacuated (days 5 through 12), the beat frequencyslowly drifts back toward its original dry value of 664 MHz, changing ata rate of about 1 MHz/d. During the second period under vacuum, from day26 to 34, the cavity shows roughly the same drying trend (about 0.7MHz/d) until reaching 664 MHz, at which point drying appears to continueonly at the much-reduced rate of 0.2 MHz/d.

The shift in beat frequency between the two cavities is as much as 13MHz. Over a more realistic range of vapor pressure, from 1650 Pa to 370Pa (corresponding to relative humidity ranging from 71% to 16% at 20°C.), the intercavity beat frequency appears to shift by as much as 5MHz. Under the assumption that all of the cavity mirrors behave in thesame manner, this would imply that the absolute shift of the shortcavity frequency is 6.3 MHz (1.3×10⁻⁸ relative error) while the longcavity shifts by 1.3 MHz (2.8×10⁻⁹). Direct measurement of the shift ofthe long cavity suggests that the actual problem is probably somewhatlarger than this value, but well below 1×10⁻⁸.

Humidity response can be highly dependent on the type of mirror coatingsused. Currently we are using mirrors with an RF magnetron sputtered SiO₂outer layer covering evaporated SiO₂/TiO₂ inner layers.

(d) Other Tests: For dry gasses we expect to achieve much loweruncertainty than when measuring moist air. As described in previouspublications [2,15], we have used our FPI refractometer to measure themolar refractivity of nitrogen and we find a result that is consistentwith other studies, within the uncertainties of our measurement and ofthe previous measurements (a few parts in 10⁸). However, this test is oflimited value for exploring the ultimate achievable performance of theFPI refractometer, because the uncertainty of this measurement isdominated by uncertainties in measuring the nitrogen pressure andtemperature. (These measurements are not needed to determine refractiveindex, but they are needed to determine the molar refractivity.) A moresensitive test (of a limited subset of all possible errors) is tocompare results between our two FPI systems when they simultaneouslymeasure the same sample of nitrogen gas. The result is independent ofuncertainty in the pressure and temperature measurement, but will exposemany other potential errors because the two systems (gas laser locked tothe long cavity and diode laser locked to a much shorter cavity) aresufficiently different that the comparison will uncover many potentialerrors, including end-effects, errors in determination of the cavityfsr, or error in accurately locking the laser to the FPI cavity. We findthat the two FPI cavities give the same answer for nitrogen refractiveindex within a few parts in 10⁹, where the residual disagreement iswithin our expected uncertainty.

The good agreement of the two FPI systems at the 10⁻⁹ level issuggestive that high accuracy can be achieved using this type of system.It is possible that relatively straightforward (but difficult)improvements in our system could achieve uncertainties well below 10⁻⁹.The high potential accuracy of the system is suggestive that refractiveindex measurements made with the system can be used to infer pressure.More precisely, from refractive index one can infer density [18], and ifeither temperature or pressure is known, the other can be calculated.Under normal laboratory conditions it is somewhat easier to measuretemperature than pressure, and therefore it is of interest to considerusing the refractive index measurements for pressure determination.Pressure transfer standards and absolute pressure standards are bothpossible.

Most straightforward is to use an FPI-based refractometer filled withnitrogen as a pressure transfer standard, calibrated against an absolutestandard. The FPI has high resolution and good long-term stability. Wecurrently achieve about 0.1 Pa useful resolution at atmosphericpressure. Note that a change in nitrogen pressure of 0.1 Pa will changethe nitrogen refractive index by 3×10⁻¹°. A temperature change of 0.3 mKwould cause this same change in refractive index, so temperature must bemeasured with this resolution, and careful monitoring of the long-termstability of the temperature sensor would be required to assure thelong-term reliability of pressure measurement. Short and long-termfrequency fluctuations of the fixed-frequency laser are also a potentialconcern if using commercial stabilized lasers, but the fluctuations arenegligible when using an iodine stabilized laser as the fixed frequencysource. If the stability of the cavity is monitored by periodicallymeasuring the laser frequency under vacuum, it should be possible tocorrect for long-term drift in cavity length.

An absolute pressure standard might also possible, based on thecalculated refractive index of helium. A relative uncertainty as low as1.8×10⁻⁶ might be achieved, limited by the current CODATA uncertainty ofthe Boltzmann constant. This corresponds to an uncertainty of 0.18 Pa atatmospheric pressure. To achieve this, it would be necessary to measurethe refractive index of helium with an uncertainty below 5×10⁻¹¹.Although there is no compelling reason that this performance could notbe achieved, it is several orders of magnitude better than what has beendemonstrated by existing refractometers.

The FPI cavity measures dispersion in helium rather than directlymeasuring refractive index. Pressure can be determined from the measureddifference between refractive indices at two widely spaced frequencies.This is again limited to a fractional uncertainty of about 2×10⁻⁶ by theBoltzmann constant. As an example of the dispersion technique, one mightimagine measuring the dispersion between 816 nm and its third harmonic,272 nm. The change in refractive index between 272 nm and 816 nm is alittle less than 1×10⁻⁶ for helium at atmospheric pressure and 20° C. Toachieve 0.2 Pa resolution, it is necessary to measure this 1×10⁻⁶ changein refractive index with a fractional uncertainty of 2×10⁻⁶,corresponding to a fractional change in laser frequency of only 2×10⁻¹².

Mirror phase shifts and the Guoy phase shift can be large relative tothe required accuracy. It is desirable to employ dielectric mirrors withgood reflectivity, but the associated phase shifts can cause problems.In theory, a cavity mirror consisting of an ideal quarter-wave stackwith the center of the reflection band at 816 nm should also providegood performance at the third harmonic. For an ideal quarter wave stackmade from nondispersive materials and operating at the center of thetransmission band, both the fundamental and third harmonic undergo a180o phase shift on reflection, and problems associated with mirrorphase shifts essentially disappear. For a real mirror made fromdispersive materials, and including possible uncertainty in the centerof the transmission band, the phase shifts might be on the order ofhundreds of kilohertz (very large relative to the desired uncertainty).However, if the phase shifts do not change appreciably when gas isadmitted to the cavity, then comparing frequencies measured under vacuumto frequencies seen when helium is present will eliminate this problemto first order. The Guoy phase shifts will not present a seriousproblem, but the mirror phase shift depends on pressure in the cavitybecause the helium refractive index changes the magnitude of thereflection at the first surface of the mirror. For ideal mirrors with180o phase shifts the problem vanishes, because the first-surfacereflection is in phase with the reflection from the bulk of the mirror.Under these circumstances the phase shift will not be affected by thechange in the magnitude of the reflection at the first surface. Onceagain, dispersion or uncertainty in the center wavelength complicatesthe picture, and it could be difficult to assure that errors at thekilohertz level are avoided; it might be necessary to use two cavitiesof different lengths to quantify this problem.

An additional problem of significance is that the dispersion of heliumis so small relative to other gasses. Impurities in the helium must bereduced below about 1 part in 107 in order to keep the fractional effecton dispersion below 2×10-6 and thus achieve 0.2 Pa uncertainty atatmospheric pressure.

While one or more embodiments have been shown and described,modifications and substitutions may be made thereto without departingfrom the spirit and scope of the invention. Accordingly, it is to beunderstood that the present invention has been described by way ofillustrations and not limitation. Embodiments herein can be usedindependently or can be combined.

All ranges disclosed herein are inclusive of the endpoints, and theendpoints are independently combinable with each other. The ranges arecontinuous and thus contain every value and subset thereof in the range.Unless otherwise stated or contextually inapplicable, all percentages,when expressing a quantity, are weight percentages. The suffix “(s)” asused herein is intended to include both the singular and the plural ofthe term that it modifies, thereby including at least one of that term(e.g., the colorant(s) includes at least one colorants). “Optional” or“optionally” means that the subsequently described event or circumstancecan or cannot occur, and that the description includes instances wherethe event occurs and instances where it does not. As used herein,“combination” is inclusive of blends, mixtures, alloys, reactionproducts, and the like.

As used herein, “a combination thereof” refers to a combinationcomprising at least one of the named constituents, components,compounds, or elements, optionally together with one or more of the sameclass of constituents, components, compounds, or elements.

All references are incorporated herein by reference.

The use of the terms “a” and “an” and “the” and similar referents in thecontext of describing the invention (especially in the context of thefollowing claims) are to be construed to cover both the singular and theplural, unless otherwise indicated herein or clearly contradicted bycontext. “Or” means “and/or.” It should further be noted that the terms“first,” “second,” “primary,” “secondary,” and the like herein do notdenote any order, quantity, or importance, but rather are used todistinguish one element from another. The modifier “about” used inconnection with a quantity is inclusive of the stated value and has themeaning dictated by the context (e.g., it includes the degree of errorassociated with measurement of the particular quantity). The conjunction“or” is used to link objects of a list or alternatives and is notdisjunctive; rather the elements can be used separately or can becombined together under appropriate circumstances.

What is claimed is:
 1. A deformometer for determining deformation of anoptical cell optic disposed on an optical cell, the deformometercomprising: an optical cell including a cell body; an entry optical celloptic disposed at an entry end of the cell body and that receivescombined light; and an exit optical cell optic disposed at an exit endof the cell body, wherein the entry optical cell optic is in opticalcommunication and optically opposes the exit optical cell optic, suchthat the exit optical cell optic receives the combined light from theentry optical cell optic, and the optical cell produces shifted combinedlight from the combined light; a first laser in optical communicationwith the entry optical cell optic and that provides first light; asecond laser in optical communication with the entry optical cell opticand that provides second light; a propagation coupler in opticalcommunication with the first laser; a beam splitter that receives thefirst light and the second light before communication into the opticalcell; a second beam splitter to receive filtered combined light andshifted combined light from the optical cell; an optical combiner thatsplits filtered light received from the second beam splitter andproduces a first cavity signal and a second cavity signal; a first lightdetector that receives the first cavity signal and produces a secondfiltered light; a second phase detector that receives the secondfiltered light from the first light detector; a second light detectorthat receives the second cavity signal from the optical combiner andproduces a first cavity signal; and a first phase detector that receivesthe first cavity signal from the second light detector.
 2. A process fordetermining deformation of an optical cell optic disposed on an opticalcell with a deformometer, the process comprising: combining first lightwith a second light; producing combined light from the first light andthe second light; receiving, by an entry optical cell optic disposed atan entry end of a cell body of a deformometer, the combined light;transmitting, from the first optical cell optic to an exit optical celloptic disposed at an exit end of the cell body, the combined light, theentry optical cell optic being in optical communication and opticallyopposing the exit optical cell optic; receiving, by the exit opticalcell optic, the combined light; producing a filtered combined light fromthe combined light transmitted from the first optical cell optic andfrom the second optical cell optic; producing, from the filteredcombined light, a first filtered light and a second filtered light; andanalyzing the first filtered light and the second filtered light todetermine the deformation of the entry optical cell optic and the exitoptical cell optic.